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Beyond Advanced · Integral Calculus

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1
Which of the following is an antiderivative of x4+1x6+1\dfrac{x^{4}+1}{x^{6}+1} on R\mathbb{R}?
Not attempted
A
tan⁡−1x+tan⁡−1 ⁣(x3)\tan^{-1}x+\tan^{-1}\!\left(x^{3}\right)
B
12tan⁡−1 ⁣(x2)+13tan⁡−1 ⁣(x3)\tfrac12\tan^{-1}\!\left(x^{2}\right)+\tfrac13\tan^{-1}\!\left(x^{3}\right)
C
tan⁡−1x+13tan⁡−1 ⁣(x3)\tan^{-1}x+\tfrac13\tan^{-1}\!\left(x^{3}\right)
D
tan⁡−1x−13tan⁡−1 ⁣(x3)\tan^{-1}x-\tfrac13\tan^{-1}\!\left(x^{3}\right)
2
Decide which statements are true.
Not attempted
A
f(x)=sgn⁡(x)f(x)=\operatorname{sgn}(x) (with sgn⁡(0)=0\operatorname{sgn}(0)=0) has no antiderivative on R\mathbb{R}.
B
f(x)=2xsin⁡1x−cos⁡1xf(x)=2x\sin\frac1x-\cos\frac1x for x≠0x\neq0 and f(0)=0f(0)=0 has an antiderivative on R\mathbb{R}, although ff is discontinuous at 00.
C
f(x)=1f(x)=1 for x≠0x\neq0 and f(0)=0f(0)=0 has an antiderivative on R\mathbb{R}.
D
Any two antiderivatives of 1x\dfrac1x on R∖{0}\mathbb{R}\setminus\{0\} differ by a constant.
3
Let f(x)=max⁡(sin⁡x,cos⁡x)f(x)=\max(\sin x,\cos x) and let FF be the antiderivative of ff on R\mathbb{R} with F(0)=1F(0)=1 (an antiderivative is differentiable, hence continuous). Find F(2π)F(2\pi), correct to two decimal places.
Not attempted
xyOπ/2π3π/22πdashed: sin x, cos x; solid: f(x)
Your answer: —
Correct answer: 3.83
4
Which of the following are antiderivatives of f(x)=x2−1(x2+1)x4+1f(x)=\dfrac{x^{2}-1}{(x^{2}+1)\sqrt{x^{4}+1}} on (0,∞)(0,\infty)?
Not attempted
A
12cos⁡−1 ⁣2 xx2+1\dfrac{1}{\sqrt2}\cos^{-1}\!\dfrac{\sqrt2\,x}{x^{2}+1}
B
−12sin⁡−1 ⁣2 xx2+1-\dfrac{1}{\sqrt2}\sin^{-1}\!\dfrac{\sqrt2\,x}{x^{2}+1}
C
12sin⁡−1 ⁣2 xx2+1\dfrac{1}{\sqrt2}\sin^{-1}\!\dfrac{\sqrt2\,x}{x^{2}+1}
D
12tan⁡−1 ⁣x2−12 x\dfrac{1}{\sqrt2}\tan^{-1}\!\dfrac{x^{2}-1}{\sqrt2\,x}
5
Let FF be the antiderivative of f(x)=1(sin⁡3x cos⁡5x)1/4f(x)=\dfrac{1}{\left(\sin^{3}x\,\cos^{5}x\right)^{1/4}} on (0,π2)\left(0,\frac{\pi}{2}\right) with F ⁣(π4)=3F\!\left(\frac{\pi}{4}\right)=3. Find F ⁣(π3)F\!\left(\frac{\pi}{3}\right) correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 3.59
6
For x>0x>0, ∫(ln⁡x−11+(ln⁡x)2)2dx\displaystyle\int\left(\frac{\ln x-1}{1+(\ln x)^{2}}\right)^{2}dx equals (up to a constant)
Not attempted
A
x1+ln⁡x\dfrac{x}{1+\ln x}
B
xln⁡x1+(ln⁡x)2\dfrac{x\ln x}{1+(\ln x)^{2}}
C
−x1+(ln⁡x)2-\dfrac{x}{1+(\ln x)^{2}}
D
x1+(ln⁡x)2\dfrac{x}{1+(\ln x)^{2}}
7
Let FF be an antiderivative of f(x)=1xf(x)=\dfrac1x on R∖{0}\mathbb{R}\setminus\{0\} with F(1)=0F(1)=0 and F(−1)=3F(-1)=3. Which statements are true?
Not attempted
A
F(e)=1F(e)=1.
B
F(−e2)=5F(-e^{2})=5.
C
F(x)+F(−x)F(x)+F(-x) is constant for x>0x>0.
D
F(x)−F(−x)=−3F(x)-F(-x)=-3 for every x>0x>0.
8
For x>2x>2, ∫dx(x+1)3/4(x−2)5/4\displaystyle\int\frac{dx}{(x+1)^{3/4}(x-2)^{5/4}} equals (up to a constant)
Not attempted
A
−43(x−2x+1)1/4-\dfrac43\left(\dfrac{x-2}{x+1}\right)^{1/4}
B
−43(x+1x−2)1/4-\dfrac43\left(\dfrac{x+1}{x-2}\right)^{1/4}
C
43(x+1x−2)1/4\dfrac43\left(\dfrac{x+1}{x-2}\right)^{1/4}
D
−43(x+1x−2)3/4-\dfrac43\left(\dfrac{x+1}{x-2}\right)^{3/4}
9
Evaluate ∫01dx(x+1+x2)3\displaystyle\int_0^{1}\frac{dx}{\left(x+\sqrt{1+x^{2}}\right)^{3}} correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.33
10
For 0<x<π20<x<\dfrac{\pi}{2}, ∫sin⁡x+cos⁡xsin⁡2x dx\displaystyle\int\frac{\sin x+\cos x}{\sqrt{\sin 2x}}\,dx equals (up to a constant)
Not attempted
A
sin⁡−1(cos⁡x−sin⁡x)\sin^{-1}(\cos x-\sin x)
B
ln⁡(sin⁡x+cos⁡x)\ln(\sin x+\cos x)
C
sin⁡−1(sin⁡x−cos⁡x)\sin^{-1}(\sin x-\cos x)
D
2sin⁡2x2\sqrt{\sin 2x}
11
Which of the following (each valid on a suitable interval, up to an additive constant) are correct?
Not attempted
A
∫dxex+e−x=tan⁡−1 ⁣(ex)\displaystyle\int\frac{dx}{e^{x}+e^{-x}}=\tan^{-1}\!\left(e^{x}\right)
B
∫dx1+ex=ln⁡(1+ex)−x\displaystyle\int\frac{dx}{1+e^{x}}=\ln\left(1+e^{x}\right)-x
C
∫dxx(1+x3)=13ln⁡1+x3x3\displaystyle\int\frac{dx}{x\left(1+x^{3}\right)}=\frac13\ln\frac{1+x^{3}}{x^{3}} for x>0x>0
D
∫1−x1+x dx=sin⁡−1x−1−x2\displaystyle\int\sqrt{\frac{1-x}{1+x}}\,dx=\sin^{-1}x-\sqrt{1-x^{2}} for ∣x∣<1|x|<1
12
Let FF be the antiderivative of f(x)=x1+x2f(x)=\dfrac{x}{1+x^{2}} on R\mathbb{R} with F(0)=0F(0)=0 (the graph of ff is shown). Which figure shows the graph of FF on [−4,4][-4,4]? Read the values carefully: ln⁡17=2.83\ln17=2.83.
Not attempted
xf(x)O−4−2240.5−0.5
A
Odd
xyO−4−2241−1
B
Even, bounded above
xyO−4−2241−1
C
Even, unbounded, with slope tending to 00
xyO−4−2241−1
D
Even, opening downward
xyO−4−2241−1
13
Let FF be the antiderivative of f(x)=1−sin⁡2xf(x)=\sqrt{1-\sin2x} on [0,π2]\left[0,\frac{\pi}{2}\right] with F(0)=0F(0)=0. Which statements are true?
Not attempted
xyOπ/4π/21
A
F ⁣(π2)=22−2F\!\left(\frac{\pi}{2}\right)=2\sqrt2-2.
B
FF is differentiable at x=π4x=\frac{\pi}{4} with F′ ⁣(π4)=0F'\!\left(\frac{\pi}{4}\right)=0.
C
F(x)=sin⁡x+cos⁡x−1F(x)=\sin x+\cos x-1 for every x∈[0,π2]x\in\left[0,\frac{\pi}{2}\right].
D
F ⁣(π2)=0F\!\left(\frac{\pi}{2}\right)=0.
14
Match each definite integral in List-I with its value in List-II. List-IList-IIP)∫01dx1+x1)2−2ln⁡2Q)∫14dxx(1+x)2)2ln⁡43R)∫0π/2sin⁡2x1+sin⁡2x dx3)ln⁡2S)∫01dxex+e−x4)tan⁡−1e−π45)π4\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \int_0^{1}\dfrac{dx}{1+\sqrt x} & \text{1)} 2-2\ln2 \\ \text{Q)} \int_1^{4}\dfrac{dx}{x\left(1+\sqrt x\right)} & \text{2)} 2\ln\dfrac43 \\ \text{R)} \int_0^{\pi/2}\dfrac{\sin2x}{1+\sin^{2}x}\,dx & \text{3)} \ln2 \\ \text{S)} \int_0^{1}\dfrac{dx}{e^{x}+e^{-x}} & \text{4)} \tan^{-1}e-\dfrac{\pi}{4} \\ & \text{5)} \dfrac{\pi}{4} \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-3, S-4
C
P-1, Q-2, R-5, S-3
D
P-3, Q-1, R-2, S-5
15
∫etan⁡−1x 1+x(1+x2)3/2 dx\displaystyle\int e^{\tan^{-1}x}\,\frac{1+x}{(1+x^{2})^{3/2}}\,dx equals (up to a constant)
Not attempted
A
x etan⁡−1x1+x2\dfrac{x\,e^{\tan^{-1}x}}{\sqrt{1+x^{2}}}
B
etan⁡−1x1+x2\dfrac{e^{\tan^{-1}x}}{\sqrt{1+x^{2}}}
C
(1+x) etan⁡−1x1+x2\dfrac{(1+x)\,e^{\tan^{-1}x}}{\sqrt{1+x^{2}}}
D
x etan⁡−1x1+x2\dfrac{x\,e^{\tan^{-1}x}}{1+x^{2}}
16
Evaluate ∫01(sin⁡−1x)2dx\displaystyle\int_0^{1}\left(\sin^{-1}x\right)^{2}dx correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.47
17
∫xcos⁡(ln⁡x) dx\displaystyle\int x\cos(\ln x)\,dx equals (up to a constant)
Not attempted
A
x25(cos⁡(ln⁡x)+2sin⁡(ln⁡x))\dfrac{x^{2}}{5}\left(\cos(\ln x)+2\sin(\ln x)\right)
B
x22(cos⁡(ln⁡x)+sin⁡(ln⁡x))\dfrac{x^{2}}{2}\left(\cos(\ln x)+\sin(\ln x)\right)
C
x25(2cos⁡(ln⁡x)−sin⁡(ln⁡x))\dfrac{x^{2}}{5}\left(2\cos(\ln x)-\sin(\ln x)\right)
D
x25(2cos⁡(ln⁡x)+sin⁡(ln⁡x))\dfrac{x^{2}}{5}\left(2\cos(\ln x)+\sin(\ln x)\right)
18
Let Jn=∫1e(ln⁡x)n dxJ_{n}=\displaystyle\int_1^{e}(\ln x)^{n}\,dx for n=0,1,2,…n=0,1,2,\dots Which statements are true?
Not attempted
A
J3=6−2eJ_{3}=6-2e.
B
J4=9e−24J_{4}=9e-24.
C
Jn<en+1J_{n}<\dfrac{e}{n+1} for every n≥1n\ge1.
D
lim⁡n→∞nJn=0\displaystyle\lim_{n\to\infty}nJ_{n}=0.
19
On (0,π)\left(0,\pi\right) the quantity xcos⁡x−sin⁡xx\cos x-\sin x never vanishes. Then ∫x2(xcos⁡x−sin⁡x)2 dx\displaystyle\int\frac{x^{2}}{(x\cos x-\sin x)^{2}}\,dx equals (up to a constant)
Not attempted
A
xsin⁡x+cos⁡xxcos⁡x+sin⁡x\dfrac{x\sin x+\cos x}{x\cos x+\sin x}
B
−xsin⁡x+cos⁡xxcos⁡x−sin⁡x-\dfrac{x\sin x+\cos x}{x\cos x-\sin x}
C
xsin⁡x+cos⁡xxcos⁡x−sin⁡x\dfrac{x\sin x+\cos x}{x\cos x-\sin x}
D
xsin⁡x−cos⁡xxcos⁡x−sin⁡x\dfrac{x\sin x-\cos x}{x\cos x-\sin x}
20
Let F(x)=∫0xt e−t dtF(x)=\displaystyle\int_0^{x}t\,e^{-t}\,dt for x≥0x\ge0 (the integrand is plotted). Which figure shows the graph of FF on [0,6.4][0,6.4]?
Not attempted
tt e^(−t)O12461/e
A
Starts with zero slope, then steepens, and levels off at 11
xyO2460.51
B
Starts with slope 11 and levels off at 11
xyO2460.51
C
Rises to a peak and then decays to 00
xyO2460.51
D
Starts at 11 and decreases to 00
xyO2460.51
21
Evaluate ∫02πe−xsin⁡2x dx\displaystyle\int_0^{2\pi}e^{-x}\sin^{2}x\,dx correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.4
22
Which of the following are antiderivatives of f(x)=x2−1x4+3x2+1f(x)=\dfrac{x^{2}-1}{x^{4}+3x^{2}+1} on (0,∞)(0,\infty)?
Not attempted
A
tan⁡−1 ⁣(x+1x)\tan^{-1}\!\left(x+\dfrac1x\right)
B
−tan⁡−1 ⁣xx2+1-\tan^{-1}\!\dfrac{x}{x^{2}+1}
C
tan⁡−1 ⁣(x−1x)\tan^{-1}\!\left(x-\dfrac1x\right)
D
12ln⁡(x2+3+1x2)\dfrac12\ln\left(x^{2}+3+\dfrac1{x^{2}}\right)
23
Evaluate ∫01dxx4+x2+1\displaystyle\int_0^{1}\frac{dx}{x^{4}+x^{2}+1} correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.73
24
For x>0x>0, ∫x+1x(1+xex) dx\displaystyle\int\frac{x+1}{x\left(1+xe^{x}\right)}\,dx equals (up to a constant)
Not attempted
A
ln⁡1+xexxex\ln\dfrac{1+xe^{x}}{xe^{x}}
B
ln⁡x1+xex\ln\dfrac{x}{1+xe^{x}}
C
ln⁡(1+xex)\ln\left(1+xe^{x}\right)
D
ln⁡xex1+xex\ln\dfrac{xe^{x}}{1+xe^{x}}
25
For x>2x>2, ∫dx(x2−1)(x2−4)\displaystyle\int\frac{dx}{(x^{2}-1)(x^{2}-4)} equals (up to a constant)
Not attempted
A
112ln⁡x−2x+2−16ln⁡x−1x+1\dfrac1{12}\ln\dfrac{x-2}{x+2}-\dfrac16\ln\dfrac{x-1}{x+1}
B
16ln⁡x−2x+2−13ln⁡x−1x+1\dfrac16\ln\dfrac{x-2}{x+2}-\dfrac13\ln\dfrac{x-1}{x+1}
C
112ln⁡x−2x+2+16ln⁡x−1x+1\dfrac1{12}\ln\dfrac{x-2}{x+2}+\dfrac16\ln\dfrac{x-1}{x+1}
D
14ln⁡x−2x+2−12ln⁡x−1x+1\dfrac14\ln\dfrac{x-2}{x+2}-\dfrac12\ln\dfrac{x-1}{x+1}
26
Let f(x)=x3+2(x−1)2(x2+1)f(x)=\dfrac{x^{3}+2}{(x-1)^{2}(x^{2}+1)}. Which statements are true?
Not attempted
A
In the partial-fraction decomposition of ff the coefficient of 1x−1\dfrac{1}{x-1} is 00.
B
∫f(x) dx=−32(x−1)+12ln⁡(x2+1)+12tan⁡−1x+C\displaystyle\int f(x)\,dx=-\frac{3}{2(x-1)}+\frac12\ln\left(x^{2}+1\right)+\frac12\tan^{-1}x+C.
C
∫23f(x) dx=34+12ln⁡52+12(tan⁡−13−tan⁡−12)\displaystyle\int_2^{3}f(x)\,dx=\frac34+\frac12\ln\frac52+\frac12\left(\tan^{-1}3-\tan^{-1}2\right).
D
ff has an antiderivative on (1,∞)(1,\infty) that is a rational function.
27
Evaluate ∫15x2+1x4+1 dx\displaystyle\int_1^{5}\frac{x^{2}+1}{x^{4}+1}\,dx correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.91
28
Match each definite integral in List-I with its value in List-II. List-IList-IIP)∫23dxx2−11)12ln⁡32Q)∫01dxx2+x+12)π33R)∫01x dx(x+1)(x2+1)3)π8−14ln⁡2S)∫01dxx3+14)13ln⁡2+π335)ln⁡43\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \int_2^{3}\dfrac{dx}{x^{2}-1} & \text{1)} \dfrac12\ln\dfrac32 \\ \text{Q)} \int_0^{1}\dfrac{dx}{x^{2}+x+1} & \text{2)} \dfrac{\pi}{3\sqrt3} \\ \text{R)} \int_0^{1}\dfrac{x\,dx}{(x+1)(x^{2}+1)} & \text{3)} \dfrac{\pi}{8}-\dfrac14\ln2 \\ \text{S)} \int_0^{1}\dfrac{dx}{x^{3}+1} & \text{4)} \dfrac13\ln2+\dfrac{\pi}{3\sqrt3} \\ & \text{5)} \ln\dfrac43 \end{array}
Not attempted
A
P-5, Q-2, R-3, S-1
B
P-1, Q-4, R-3, S-2
C
P-1, Q-2, R-3, S-4
D
P-2, Q-1, R-5, S-4
29
Let f(x)=12+cos⁡xf(x)=\dfrac{1}{2+\cos x}. A student uses t=tan⁡x2t=\tan\frac x2 to get the antiderivative G(x)=23tan⁡−1 ⁣(13tan⁡x2)G(x)=\dfrac{2}{\sqrt3}\tan^{-1}\!\left(\dfrac{1}{\sqrt3}\tan\dfrac x2\right) and computes G(2π)−G(0)G(2\pi)-G(0). The true value of ∫02πf(x) dx\displaystyle\int_0^{2\pi}f(x)\,dx, correct to two decimal places, is
Not attempted
xyOπ2π1/31
Your answer: —
Correct answer: 3.63
30
For x>−1x>-1, ∫dx(x+1)x2+x+1\displaystyle\int\frac{dx}{(x+1)\sqrt{x^{2}+x+1}} equals (up to a constant)
Not attempted
A
ln⁡1−x+2x2+x+1x+1\ln\dfrac{1-x+2\sqrt{x^{2}+x+1}}{x+1}
B
−ln⁡x+1+2x2+x+1x+1-\ln\dfrac{x+1+2\sqrt{x^{2}+x+1}}{x+1}
C
−ln⁡1−x+2x2+x+1x+1-\ln\dfrac{1-x+2\sqrt{x^{2}+x+1}}{x+1}
D
−ln⁡1−x+2x2+x+1x−1-\ln\dfrac{1-x+2\sqrt{x^{2}+x+1}}{x-1}
31
Evaluate ∫0π/22sin⁡x+3cos⁡x3sin⁡x+4cos⁡x dx\displaystyle\int_0^{\pi/2}\frac{2\sin x+3\cos x}{3\sin x+4\cos x}\,dx correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.12
32
Which of the following are antiderivatives of f(x)=1(x2+1)x2+2f(x)=\dfrac{1}{(x^{2}+1)\sqrt{x^{2}+2}} on (0,∞)(0,\infty)?
Not attempted
A
tan⁡−1 ⁣xx2+2\tan^{-1}\!\dfrac{x}{\sqrt{x^{2}+2}}
B
sin⁡−1 ⁣x2(x2+1)\sin^{-1}\!\dfrac{x}{\sqrt{2(x^{2}+1)}}
C
tan⁡−1 ⁣xx2+1\tan^{-1}\!\dfrac{x}{\sqrt{x^{2}+1}}
D
−tan⁡−1 ⁣x2+2x-\tan^{-1}\!\dfrac{\sqrt{x^{2}+2}}{x}
33
Which of the following are correct?
Not attempted
A
∫024x−x2 dx=π\displaystyle\int_0^{2}\sqrt{4x-x^{2}}\,dx=\pi.
B
∫01x1−x dx=π2\displaystyle\int_0^{1}\sqrt{\frac{x}{1-x}}\,dx=\frac\pi2.
C
∫01dxx2+2x+2=ln⁡2+51+2\displaystyle\int_0^{1}\frac{dx}{\sqrt{x^{2}+2x+2}}=\ln\frac{2+\sqrt5}{1+\sqrt2}.
D
∫0π/2dx2+cos⁡x=π6\displaystyle\int_0^{\pi/2}\frac{dx}{2+\cos x}=\frac\pi6.
34
For 0<x<10<x<1, ∫x1−x3 dx\displaystyle\int\sqrt{\frac{x}{1-x^{3}}}\,dx equals (up to a constant)
Not attempted
A
sin⁡−1 ⁣(x3/2)\sin^{-1}\!\left(x^{3/2}\right)
B
23sin⁡−1 ⁣(x3/2)\dfrac23\sin^{-1}\!\left(x^{3/2}\right)
C
23sin⁡−1 ⁣(x3)\dfrac23\sin^{-1}\!\left(x^{3}\right)
D
23tan⁡−1 ⁣(x3/2)\dfrac23\tan^{-1}\!\left(x^{3/2}\right)
35
Evaluate ∫0πdx9sin⁡2x+16cos⁡2x\displaystyle\int_0^{\pi}\frac{dx}{9\sin^{2}x+16\cos^{2}x} correct to two decimal places.
Not attempted
xyOπ/2π
Your answer: —
Correct answer: 0.26
36
Evaluate lim⁡n→∞1n[(n+1)(n+2)⋯(2n)]1/n\displaystyle\lim_{n\to\infty}\frac1n\Big[(n+1)(n+2)\cdots(2n)\Big]^{1/n} correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.47
37
lim⁡n→∞∑k=1nkn2+nk+k2\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^{2}+nk+k^{2}} equals
Not attempted
A
12ln⁡3−π63\dfrac12\ln3-\dfrac{\pi}{6\sqrt3}
B
12ln⁡3+π63\dfrac12\ln3+\dfrac{\pi}{6\sqrt3}
C
ln⁡3−π33\ln3-\dfrac{\pi}{3\sqrt3}
D
12ln⁡3−π33\dfrac12\ln3-\dfrac{\pi}{3\sqrt3}
38
Which statements are true?
Not attempted
A
The Riemann integral ∫01sin⁡1x dx\displaystyle\int_0^{1}\sin\frac1x\,dx exists (define the integrand to be 00 at x=0x=0).
B
If ff is continuous on [a,b][a,b] and ∫abf(x) dx=0\displaystyle\int_a^{b}f(x)\,dx=0, then f(c)=0f(c)=0 for some c∈(a,b)c\in(a,b).
C
If ff is Riemann integrable on [a,b][a,b] and ∫abf(x)2dx=0\displaystyle\int_a^{b}f(x)^{2}dx=0, then f(x)=0f(x)=0 for every x∈[a,b]x\in[a,b].
D
If f(x)=⌊x⌋f(x)=\lfloor x\rfloor on [0,3][0,3], then F(x)=∫0xf(t) dtF(x)=\displaystyle\int_0^{x}f(t)\,dt is differentiable at x=1x=1.
39
Let Sn=∑k=1n2k⌊k ⌋S_n=\displaystyle\sum_{k=1}^{n^{2}}k\left\lfloor\sqrt k\,\right\rfloor. Find lim⁡n→∞Snn5\displaystyle\lim_{n\to\infty}\frac{S_n}{n^{5}} correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.4
40
Let F(x)=∫0x⌊t⌋ dtF(x)=\displaystyle\int_0^{x}\lfloor t\rfloor\,dt for 0≤x≤3.50\le x\le3.5, where ⌊⋅⌋\lfloor\cdot\rfloor is the greatest-integer function (the integrand is plotted). Which figure shows the graph of FF?
Not attempted
tf(t)O123123
A
A staircase of horizontal steps
xyO123136
B
Continuous and piecewise linear, flat at the start, slopes 0,1,2,30,1,2,3
xyO123136
C
A smooth parabola
xyO123136
D
Continuous and piecewise linear, slopes 1,2,31,2,3 from the start
xyO123136
41
lim⁡n→∞n∫01xn1+x2 dx\displaystyle\lim_{n\to\infty}n\int_0^{1}\frac{x^{n}}{1+x^{2}}\,dx equals
Not attempted
A
00
B
12\dfrac12
C
14\dfrac14
D
11
42
Match each limit (as n→∞n\to\infty) in List-I with its value in List-II. List-IList-IIP)∑k=1nnn2+k21)π4Q)∑k=0n−11n2−k22)π2R)(n!)1/nn3)1eS)∑k=1nkn2+k24)12ln⁡25)4e\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \displaystyle\sum_{k=1}^{n}\frac{n}{n^{2}+k^{2}} & \text{1)} \dfrac\pi4 \\ \text{Q)} \displaystyle\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^{2}-k^{2}}} & \text{2)} \dfrac\pi2 \\ \text{R)} \dfrac{(n!)^{1/n}}{n} & \text{3)} \dfrac1e \\ \text{S)} \displaystyle\sum_{k=1}^{n}\frac{k}{n^{2}+k^{2}} & \text{4)} \dfrac12\ln2 \\ & \text{5)} \dfrac4e \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-5, S-3
C
P-4, Q-2, R-1, S-3
D
P-1, Q-2, R-3, S-4
43
Evaluate ∫02πxsin⁡4x dx\displaystyle\int_0^{2\pi}x\sin^{4}x\,dx correct to two decimal places.
Not attempted
xyOπ2π
Your answer: —
Correct answer: 7.4
44
∫0π/2ln⁡(tan⁡x+cot⁡x)dx\displaystyle\int_0^{\pi/2}\ln\left(\tan x+\cot x\right)dx equals
Not attempted
A
πln⁡2\pi\ln2
B
π2ln⁡2\dfrac{\pi}{2}\ln2
C
2πln⁡22\pi\ln2
D
00
45
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be continuous, odd, and periodic with period 33, and let g(x)=∫0xf(t) dtg(x)=\displaystyle\int_0^{x}f(t)\,dt. Which statements are true?
Not attempted
A
f ⁣(32)=0f\!\left(\dfrac32\right)=0.
B
∫03f(x) dx=0\displaystyle\int_0^{3}f(x)\,dx=0.
C
gg is periodic with period 33.
D
gg is an even function.
46
∫−π/2π/2sin⁡4x1+e xcos⁡x dx\displaystyle\int_{-\pi/2}^{\pi/2}\frac{\sin^{4}x}{1+e^{\,x\cos x}}\,dx equals
Not attempted
A
3π8\dfrac{3\pi}{8}
B
3π32\dfrac{3\pi}{32}
C
3π16\dfrac{3\pi}{16}
D
π4\dfrac{\pi}{4}
47
Evaluate ∫0πx1+sin⁡x dx\displaystyle\int_0^{\pi}\frac{x}{1+\sin x}\,dx correct to two decimal places.
Not attempted
xyOππ/2
Your answer: —
Correct answer: 3.14
48
Let ff be continuous on [0,1][0,1] with f(x)=f(1−x)f(x)=f(1-x) for all xx. Which statements are true?
Not attempted
A
∫01x f(x) dx=12∫01f(x) dx\displaystyle\int_0^{1}x\,f(x)\,dx=\frac12\int_0^{1}f(x)\,dx.
B
∫01x2f(x) dx=13∫01f(x) dx\displaystyle\int_0^{1}x^{2}f(x)\,dx=\frac13\int_0^{1}f(x)\,dx.
C
∫01(2x−1)3f(x) dx=0\displaystyle\int_0^{1}(2x-1)^{3}f(x)\,dx=0.
D
∫01f(x)cos⁡(πx) dx=0\displaystyle\int_0^{1}f(x)\cos(\pi x)\,dx=0.
49
∫0π/2xsin⁡x+cos⁡x dx\displaystyle\int_0^{\pi/2}\frac{x}{\sin x+\cos x}\,dx equals
Not attempted
A
π2ln⁡(1+2)\dfrac{\pi}{\sqrt2}\ln\left(1+\sqrt2\right)
B
π4ln⁡(1+2)\dfrac{\pi}{4}\ln\left(1+\sqrt2\right)
C
π22ln⁡(2−1)\dfrac{\pi}{2\sqrt2}\ln\left(\sqrt2-1\right)
D
π22ln⁡(1+2)\dfrac{\pi}{2\sqrt2}\ln\left(1+\sqrt2\right)
50
The graph of a function ff on [0,4][0,4] is shown: f(t)=2f(t)=2 for 0≤t≤10\le t\le1, ff decreases linearly from 22 to −2-2 on 1≤t≤31\le t\le3, and f(t)=−2f(t)=-2 for 3≤t≤43\le t\le4. Which figure shows the graph of F(x)=∫0xf(t) dtF(x)=\int_0^{x}f(t)\,dt on [0,4][0,4]?
Not attempted
tf(t)O12342−2
A
Falls steadily, rounds off to a minimum, rises, then ascends along a straight line
xyO12343−36
B
Increases throughout, never decreasing
xyO12343−36
C
Rises steadily, rounds off to a maximum, falls, then descends along a straight line
xyO12343−36
D
Constant, then falls in a straight line, then constant
xyO12343−36
51
lim⁡x→0+1x3∫0x2sin⁡t  dt\displaystyle\lim_{x\to0^{+}}\frac{1}{x^{3}}\int_0^{x^{2}}\sin\sqrt t\;dt equals
Not attempted
A
23\dfrac23
B
13\dfrac13
C
11
D
00
52
A differentiable function f:R→Rf:\mathbb{R}\to\mathbb{R} satisfies f(x)=2x+∫0x(x−t) f(t) dtf(x)=2x+\displaystyle\int_0^{x}(x-t)\,f(t)\,dt for all xx. Find f(1)f(1) correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.35
53
Let F(x)=∫0x(t−1)(t−2)2e−t dtF(x)=\displaystyle\int_0^{x}(t-1)(t-2)^{2}e^{-t}\,dt for x∈Rx\in\mathbb{R} (the graph of the integrand is shown). Which statements are true?
Not attempted
tf(t)O124
A
FF has a local extremum at x=2x=2.
B
The minimum value of FF on R\mathbb{R} is −3e-\dfrac3e.
C
F(x)<0F(x)<0 for all x>0x>0, and F(x)→0F(x)\to0 as x→∞x\to\infty.
D
FF is bounded on R\mathbb{R}.
54
Let φ(x)=∫0x(x−t)2ln⁡(1+t2)dt\varphi(x)=\displaystyle\int_0^{x}(x-t)^{2}\ln\left(1+t^{2}\right)dt. Then φ′′′(1)+φ′′(1)\varphi'''(1)+\varphi''(1) equals
Not attempted
A
2ln⁡22\ln2
B
4ln⁡2+π−44\ln2+\pi-4
C
4ln⁡2−π−44\ln2-\pi-4
D
2ln⁡2+π−42\ln2+\pi-4
55
Let f(x)=x+exf(x)=x+e^{x} for x∈Rx\in\mathbb{R}; it is strictly increasing, so it has an inverse gg (the figure shows x=g(y)x=g(y) for y∈[1,e+1]y\in[1,e+1]). Find ∫1e+1g(y) dy\displaystyle\int_1^{e+1}g(y)\,dy correct to two decimal places.
Not attempted
y1e+11
Your answer: —
Correct answer: 1.5
56
Match each limit in List-I with its value in List-II. List-IList-IIP)lim⁡x→01x4∫0x(t−sin⁡t) dt1)0Q)lim⁡x→∞1x∫0x∣sin⁡t∣ dt2)124R)lim⁡x→∞(∫0xet2dt)2∫0xe2t2dt3)2πS)lim⁡x→01x2∫0xln⁡(1+t) dt4)125)1\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \displaystyle\lim_{x\to0}\frac{1}{x^{4}}\int_0^{x}(t-\sin t)\,dt & \text{1)} 0 \\ \text{Q)} \displaystyle\lim_{x\to\infty}\frac1x\int_0^{x}|\sin t|\,dt & \text{2)} \dfrac1{24} \\ \text{R)} \displaystyle\lim_{x\to\infty}\frac{\left(\int_0^{x}e^{t^{2}}dt\right)^{2}}{\int_0^{x}e^{2t^{2}}dt} & \text{3)} \dfrac2\pi \\ \text{S)} \displaystyle\lim_{x\to0}\frac{1}{x^{2}}\int_0^{x}\ln(1+t)\,dt & \text{4)} \dfrac12 \\ & \text{5)} 1 \end{array}
Not attempted
A
P-2, Q-3, R-1, S-4
B
P-3, Q-2, R-1, S-4
C
P-2, Q-3, R-5, S-1
D
P-1, Q-2, R-3, S-4
57
Find the total area of the region(s) enclosed between the curves y=sin⁡xy=\sin x and y=sin⁡2xy=\sin2x for 0≤x≤π0\le x\le\pi (shaded in the figure).
Not attempted
xyOπ/3πsolid: sin x; dashed: sin 2x
A
52\dfrac52
B
22
C
94\dfrac94
D
33
58
The tangent to the curve y=ln⁡xy=\ln x is drawn from the origin. Find the area of the region bounded by the curve, this tangent and the xx-axis (shaded in the figure), correct to two decimal places.
Not attempted
xyO(e, 1)(1, 0)y = ln x and its tangent from O
Your answer: —
Correct answer: 0.36
59
For an integer n≥2n\ge2 let AnA_n be the area enclosed between the curves y=xny=x^{n} and y=x1/ny=x^{1/n} for x≥0x\ge0 (the figure shows n=3n=3). Which statements are true?
Not attempted
xyO11
A
An=n−1n+1A_n=\dfrac{n-1}{n+1}.
B
AnA_n equals twice the area enclosed between y=xny=x^{n} and the line y=xy=x.
C
AnA_n decreases as nn increases.
D
An>12A_n>\dfrac12 for every n≥4n\ge4.
60
Find the total area of the region(s) bounded by the curve y=∣x2−2x∣y=\left|x^{2}-2x\right| and the line y=xy=x (shaded in the figure).
Not attempted
xyO12323y = |x² − 2x|
A
76\dfrac{7}{6}
B
32\dfrac{3}{2}
C
136\dfrac{13}{6}
D
83\dfrac{8}{3}
61
Find the area of the region lying inside the circle x2+y2=2x^{2}+y^{2}=2 and above the parabola y=x2y=x^{2} (shaded in the figure).
Not attempted
xyO(1, 1)(−1, 1)
A
π2−13\dfrac\pi2-\dfrac13
B
π−23\pi-\dfrac23
C
π2+23\dfrac{\pi}{2}+\dfrac23
D
π2+13\dfrac{\pi}{2}+\dfrac13
62
The line y=mxy=mx (with 0<m<20<m<2) divides the region bounded by the parabola y=2x−x2y=2x-x^{2} and the xx-axis into two parts of equal area. Find mm correct to two decimal places. (Use 41/3=1.58744^{1/3}=1.5874.)
Not attempted
xyO211y = 2x − x² and y = mx
Your answer: —
Correct answer: 0.41
63
Match each region in List-I with its area in List-II. List-IList-IIP)bounded by y=x2 and y=x1)13Q)bounded by y=sin⁡x, y=cos⁡x, x=0, x=π22)2(2−1)R)bounded by y=ln⁡x, the x-axis and x=e3)1S)bounded by y=ex, y=e−x and x=14)e+1e−25)12\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{bounded by } y=x^{2} \text{ and } y=\sqrt x & \text{1)} \dfrac13 \\ \text{Q)} \text{bounded by } y=\sin x,\ y=\cos x,\ x=0,\ x=\dfrac\pi2 & \text{2)} 2(\sqrt2-1) \\ \text{R)} \text{bounded by } y=\ln x,\ \text{the } x\text{-axis and } x=e & \text{3)} 1 \\ \text{S)} \text{bounded by } y=e^{x},\ y=e^{-x} \text{ and } x=1 & \text{4)} e+\dfrac1e-2 \\ & \text{5)} \dfrac12 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-3, S-4
C
P-1, Q-5, R-3, S-2
D
P-3, Q-2, R-1, S-5
64
Evaluate 1000∫0πsin⁡6x cos⁡6x dx1000\displaystyle\int_0^{\pi}\sin^{6}x\,\cos^{6}x\,dx correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 15.34
65
Let In=∫0π/4tan⁡nx dxI_n=\displaystyle\int_0^{\pi/4}\tan^{n}x\,dx. Then I10+2I8+I6I_{10}+2I_{8}+I_{6} equals
Not attempted
A
1663\dfrac{16}{63}
B
724\dfrac{7}{24}
C
29\dfrac{2}{9}
D
1235\dfrac{12}{35}
66
Let Jn=∫0π/2sin⁡nx dxJ_{n}=\displaystyle\int_0^{\pi/2}\sin^{n}x\,dx. Which statements are true?
Not attempted
A
nJnJn−1=π2nJ_{n}J_{n-1}=\dfrac\pi2 for every n≥1n\ge1.
B
J6=5π32J_{6}=\dfrac{5\pi}{32}.
C
J7=815J_{7}=\dfrac{8}{15}.
D
lim⁡n→∞n Jn=π2\displaystyle\lim_{n\to\infty}\sqrt n\,J_{n}=\sqrt{\dfrac\pi2}.
67
∫02π⌊2sin⁡x⌋dx\displaystyle\int_0^{2\pi}\left\lfloor 2\sin x\right\rfloor dx equals (here ⌊⋅⌋\lfloor\cdot\rfloor is the greatest-integer function)
Not attempted
xyOπ2π2−21−1steps: y = ⌊2 sin x⌋
A
π\pi
B
−π-\pi
C
−2π-2\pi
D
00
68
For real xx let G(x)=∫03∣t3−x∣dtG(x)=\displaystyle\int_0^{3}\left|t^{3}-x\right|dt. Find the minimum value of GG correct to two decimal places.
Not attempted
tyOt = 3/2327
Your answer: —
Correct answer: 17.72
69
∫01(1−x2)5dx\displaystyle\int_0^{1}\left(1-x^{2}\right)^{5}dx equals
Not attempted
A
128693\dfrac{128}{693}
B
63π512\dfrac{63\pi}{512}
C
256693\dfrac{256}{693}
D
16\dfrac16
70
Match each integral in List-I with its value in List-II. List-IList-IIP)∫0π/2sin⁡5x dx1)815Q)∫0πcos⁡4x dx2)3π8R)∫0π/2sin⁡3xcos⁡2x dx3)215S)∫02πsin⁡4x dx4)3π45)1635\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \displaystyle\int_0^{\pi/2}\sin^{5}x\,dx & \text{1)} \dfrac{8}{15} \\ \text{Q)} \displaystyle\int_0^{\pi}\cos^{4}x\,dx & \text{2)} \dfrac{3\pi}{8} \\ \text{R)} \displaystyle\int_0^{\pi/2}\sin^{3}x\cos^{2}x\,dx & \text{3)} \dfrac{2}{15} \\ \text{S)} \displaystyle\int_0^{2\pi}\sin^{4}x\,dx & \text{4)} \dfrac{3\pi}{4} \\ & \text{5)} \dfrac{16}{35} \end{array}
Not attempted
A
P-5, Q-2, R-1, S-4
B
P-1, Q-4, R-3, S-2
C
P-3, Q-2, R-1, S-5
D
P-1, Q-2, R-3, S-4
Reduce each integral to Wallis' form ∫0π/2\int_0^{\pi/2}; remember a factor π2\frac\pi2 only appears when both powers are even.
P: odd power, 4⋅25⋅3⋅1=815\frac{4\cdot2}{5\cdot3\cdot1}=\frac8{15}. Q: cos⁡4\cos^{4} is symmetric about π2\frac\pi2, so the integral is 2⋅34⋅12⋅π2=3π82\cdot\frac{3}{4}\cdot\frac12\cdot\frac\pi2=\frac{3\pi}8. R: (3−1)!! (2−1)!!5!!=2⋅15⋅3⋅1=215\frac{(3-1)!!\,(2-1)!!}{5!!}=\frac{2\cdot1}{5\cdot3\cdot1}=\frac2{15} (one odd power, so no π\pi). S: four quarter-periods of ∫0π/2sin⁡4=3π16\int_0^{\pi/2}\sin^{4}=\frac{3\pi}{16}, so 3π4\frac{3\pi}4. The value 1635\frac{16}{35} is ∫0π/2sin⁡7x dx\int_0^{\pi/2}\sin^{7}x\,dx and is a distractor.