Question 1

Find the integral using an appropriate trigonometric substitution. $\int \frac{x}{\sqrt{4-x^{2}}} d x$

Accepted Answer

A)  $-\sqrt{4-x^{2}}+C$ 
B)  $\sqrt{2-x}+C$ 
C)  $\sqrt{4-x^{2}}+C$ 
D)  $-\sqrt{2-x}+C$ 
A)  $-\sqrt{4-x^{2}}+C$ 
B)  $\sqrt{2-x}+C$ 
C)  $\sqrt{4-x^{2}}+C$ 
D)  $-\sqrt{2-x}+C$

Question 2

Evaluate the indefinite integral. $\int x \cos 9 x d x$

Accepted Answer

A)  $\frac{1}{81} \sin 9 x+\frac{x}{9} \cos 9 x+C$ 
B)  $\frac{1}{81} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
C)  $\frac{1}{9} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
D)  $\frac{x}{81} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
E)  $\text { None of these }$ 
A)  $\frac{1}{81} \sin 9 x+\frac{x}{9} \cos 9 x+C$ 
B)  $\frac{1}{81} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
C)  $\frac{1}{9} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
D)  $\frac{x}{81} \cos 9 x+\frac{x}{9} \sin 9 x+C$ 
E)  $\text { None of these }$

Question 3

Find a bound on the error in approximating the integral $\int_{1}^{9} \ln x^{9}$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n=10$ subintervals.

Accepted Answer

The answer of Find a bound on the error in...

Question 4

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-27}^{8} \frac{1}{\sqrt[3]{x}} d x$

Accepted Answer

A)  $15$ 
B)  $-\frac{15}{2}$ 
C)  Diverges 
D)  $\frac{15}{2}$ 
A)  $15$ 
B)  $-\frac{15}{2}$ 
C)  Diverges 
D)  $\frac{15}{2}$

Question 5

Evaluate the integral. $\int_{x / 3}^{x / 2} 2 \cot ^{2} x d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{x / 3}^{x /...

Question 6

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{2} \frac{d x}{\sqrt{x^{3}+4}} ; \quad n=6$

Accepted Answer

A)  0.8528 
B)  0.9842 
C)  0.4921 
D)  0.4695 
A)  0.8528 
B)  0.9842 
C)  0.4921 
D)  0.4695

Question 7

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-\infty}^{\infty} \frac{3 e^{x}}{3+e^{2 x}} d x$

Accepted Answer

A)  $\frac{\pi \sqrt{3}}{2}$ 
B)  $\sqrt{3}$ 
C)  $\frac{\pi \sqrt{3}}{3}$ 
D)  Diverges 
A)  $\frac{\pi \sqrt{3}}{2}$ 
B)  $\sqrt{3}$ 
C)  $\frac{\pi \sqrt{3}}{3}$ 
D)  Diverges

Question 8

The region under the curve $y=2 \sin ^{2} x$ , $0 \leq x \leq \pi$ is rotated about the x-axis. Find the volume of the resulting solid.

Accepted Answer

A)  $\frac{3}{4}$ $\pi^{2}$ 
B)  $\frac{2}{\pi}$ 
C)  $\frac{6}{\pi^{2}}$ 
D)  $\frac{\pi}{6}$ 
E)  $\frac{\pi^{2}}{6}$ 
A)  $\frac{3}{4}$ $\pi^{2}$ 
B)  $\frac{2}{\pi}$ 
C)  $\frac{6}{\pi^{2}}$ 
D)  $\frac{\pi}{6}$ 
E)  $\frac{\pi^{2}}{6}$

Question 9

Find the integral. $\int \frac{3 x-3}{x^{2}-x-2} d x$

Accepted Answer

A)  $\ln \left|(x+2)(x-1)^{2}\right|+C$ 
B)  $\ln \left|(x-2)(x+1)^{2}\right|+C$ 
C)  $\ln \left|(x+1)(x-2)^{2}\right|+C$ 
D)  $\ln \left|(x-1)(x+2)^{2}\right|+C$ 
A)  $\ln \left|(x+2)(x-1)^{2}\right|+C$ 
B)  $\ln \left|(x-2)(x+1)^{2}\right|+C$ 
C)  $\ln \left|(x+1)(x-2)^{2}\right|+C$ 
D)  $\ln \left|(x-1)(x+2)^{2}\right|+C$

Question 10

Find the area bounded by the curves $y=3 \cos x$ and $y=3 \cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Accepted Answer

The answer of Find the area bounded by the curves...

Question 11

Find the area bounded by the curves $y=\cos x$ and $y=\cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Accepted Answer

A)  $\frac{1}{4}$ 
B)  $4$ 
C)  $\frac{\sqrt{2}}{2}$ 
D)  $\frac{1}{2}$ 
E)  $\text { none of these }$ 
A)  $\frac{1}{4}$ 
B)  $4$ 
C)  $\frac{\sqrt{2}}{2}$ 
D)  $\frac{1}{2}$ 
E)  $\text { none of these }$

Question 12

Determine whether the integral converges or diverges. If it converges, find its value. $\int_{1}^{\infty} \frac{d x}{x^{5} \ln x}$

Accepted Answer

The answer of Determine whether the integral converges or diverges....

Question 13

Evaluate the integral. $\int \frac{x^{2}}{\left(16-x^{2}\right)^{3 / 2}} d x$

Accepted Answer

The answer of Evaluate the integral. \(\int \frac{x^{2}}{\left(16-x^{2}\right)^{3 / 2}}...

Question 14

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{2 \pi}^{\infty} \cos x d x$

Accepted Answer

A)  Diverges 
B)  0 
C)  2 
D)  4 
A)  Diverges 
B)  0 
C)  2 
D)  4

Question 15

A corporation is building a complex of homes, offices, stores, schools, and churches in a rural community. As a result of this development, the planners have estimated that the community's population (in thousands) t years from now will be given by $P(t)=\frac{3 t^{2}+130 t+315}{t^{2}+6 t+45}$ .
What will the average population of the community be over the next 10 years?

Accepted Answer

The answer of A corporation is building a complex of...

Question 16

A torus is generated by rotating the circle $x^{2}+(y-6)^{2}=25$ about the x-axis. Find the volume enclosed by the torus. Round the answer to the nearest hundredth.

Accepted Answer

The answer of A torus is generated by rotating the...

Question 17

Find the volume obtained by rotating the region bounded by the given curves about $y=-1$ . $y=\sin x, x=0, x=\pi, y=0$

Accepted Answer

The answer of Find the volume obtained by rotating the...

Question 18

Find the integral. $\int \tan ^{2} x \sec ^{6} x d x$

Accepted Answer

A)  $\frac{1}{7} \tan ^{7} x+\frac{2}{5} \tan ^{5} x+\frac{1}{3} \tan ^{3} x+C$ 
B)  $\frac{1}{5} \tan ^{5} x+\frac{2}{3} \tan ^{3} x+\tan x+C$ 
C)  $\frac{1}{5} \tan ^{5} x+\frac{2}{3} \tan ^{3} x-\tan x+C$ 
D)  $\frac{1}{7} \tan ^{7} x+\frac{2}{5} \tan ^{5} x-\frac{1}{3} \tan ^{3} x+C$ 
A)  $\frac{1}{7} \tan ^{7} x+\frac{2}{5} \tan ^{5} x+\frac{1}{3} \tan ^{3} x+C$ 
B)  $\frac{1}{5} \tan ^{5} x+\frac{2}{3} \tan ^{3} x+\tan x+C$ 
C)  $\frac{1}{5} \tan ^{5} x+\frac{2}{3} \tan ^{3} x-\tan x+C$ 
D)  $\frac{1}{7} \tan ^{7} x+\frac{2}{5} \tan ^{5} x-\frac{1}{3} \tan ^{3} x+C$

Question 19

Find the volume of the resulting solid if the region under the curve $y=\frac{1}{\left(x^{2}+3 x+2\right)}$ from $x=0$ to $x=1$ is rotated about the x-axis. Round your answer to four decimal places.

Accepted Answer

The answer of Find the volume of the resulting solid...

Question 20

Evaluate the integral. $\int_{0}^{\pi / 2} 5 \sqrt{1-\cos 4 \theta} d \theta$

Accepted Answer

A)  $\frac{\sqrt{2}}{3}$ 
B)  $3 \sqrt{3}$ 
C)  $2 \sqrt{3}$ 
D)  $2 \sqrt{2}$ 
E)  $5 \sqrt{2}$ 
A)  $\frac{\sqrt{2}}{3}$ 
B)  $3 \sqrt{3}$ 
C)  $2 \sqrt{3}$ 
D)  $2 \sqrt{2}$ 
E)  $5 \sqrt{2}$

Find the integral using an appropriate trigonometric substitution. $\int \frac{x}{\sqrt{4-x^{2}}} d x$

Evaluate the indefinite integral. $\int x \cos 9 x d x$

Find a bound on the error in approximating the integral $\int_{1}^{9} \ln x^{9}$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n=10$ subintervals.

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-27}^{8} \frac{1}{\sqrt[3]{x}} d x$

Evaluate the integral. $\int_{x / 3}^{x / 2} 2 \cot ^{2} x d x$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{2} \frac{d x}{\sqrt{x^{3}+4}} ; \quad n=6$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-\infty}^{\infty} \frac{3 e^{x}}{3+e^{2 x}} d x$

The region under the curve $y=2 \sin ^{2} x$ , $0 \leq x \leq \pi$ is rotated about the x-axis. Find the volume of the resulting solid.

Find the integral. $\int \frac{3 x-3}{x^{2}-x-2} d x$

Find the area bounded by the curves $y=3 \cos x$ and $y=3 \cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Find the area bounded by the curves $y=\cos x$ and $y=\cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Determine whether the integral converges or diverges. If it converges, find its value. $\int_{1}^{\infty} \frac{d x}{x^{5} \ln x}$

Evaluate the integral. $\int \frac{x^{2}}{\left(16-x^{2}\right)^{3 / 2}} d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{2 \pi}^{\infty} \cos x d x$

A torus is generated by rotating the circle $x^{2}+(y-6)^{2}=25$ about the x-axis. Find the volume enclosed by the torus. Round the answer to the nearest hundredth.

Find the volume obtained by rotating the region bounded by the given curves about $y=-1$ . $y=\sin x, x=0, x=\pi, y=0$

Find the integral. $\int \tan ^{2} x \sec ^{6} x d x$

Find the volume of the resulting solid if the region under the curve $y=\frac{1}{\left(x^{2}+3 x+2\right)}$ from $x=0$ to $x=1$ is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. $\int_{0}^{\pi / 2} 5 \sqrt{1-\cos 4 \theta} d \theta$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 7: Techniques of Integration

Find the integral using an appropriate trigonometric substitution. ∫x4−x2dx\int \frac{x}{\sqrt{4-x^{2}}} d x∫4−x2​x​dx

Evaluate the indefinite integral. ∫xcos⁡9xdx\int x \cos 9 x d x∫xcos9xdx

Find a bound on the error in approximating the integral ∫19ln⁡x9\int_{1}^{9} \ln x^{9}∫19​lnx9 using (a) the Trapezoidal Rule and (b) Simpson's Rule with n=10n=10n=10 subintervals.

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫−2781x3dx\int_{-27}^{8} \frac{1}{\sqrt[3]{x}} d x∫−278​3x​1​dx

Evaluate the integral. ∫x/3x/22cot⁡2xdx\int_{x / 3}^{x / 2} 2 \cot ^{2} x d x∫x/3x/2​2cot2xdx

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. ∫02dxx3+4;n=6\int_{0}^{2} \frac{d x}{\sqrt{x^{3}+4}} ; \quad n=6∫02​x3+4​dx​;n=6

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫−∞∞3ex3+e2xdx\int_{-\infty}^{\infty} \frac{3 e^{x}}{3+e^{2 x}} d x∫−∞∞​3+e2x3ex​dx

The region under the curve y=2sin⁡2xy=2 \sin ^{2} xy=2sin2x , 0≤x≤π0 \leq x \leq \pi0≤x≤π is rotated about the x-axis. Find the volume of the resulting solid.

Find the integral. ∫3x−3x2−x−2dx\int \frac{3 x-3}{x^{2}-x-2} d x∫x2−x−23x−3​dx

Find the area bounded by the curves y=3cos⁡xy=3 \cos xy=3cosx and y=3cos⁡2xy=3 \cos ^{2} xy=3cos2x between x=0x=0x=0 and x=π2x=\frac{\pi}{2}x=2π​ .

Find the area bounded by the curves y=cos⁡xy=\cos xy=cosx and y=cos⁡2xy=\cos ^{2} xy=cos2x between x=0x=0x=0 and x=π2x=\frac{\pi}{2}x=2π​ .

Determine whether the integral converges or diverges. If it converges, find its value. ∫1∞dxx5ln⁡x\int_{1}^{\infty} \frac{d x}{x^{5} \ln x}∫1∞​x5lnxdx​

Evaluate the integral. ∫x2(16−x2)3/2dx\int \frac{x^{2}}{\left(16-x^{2}\right)^{3 / 2}} d x∫(16−x2)3/2x2​dx

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫2π∞cos⁡xdx\int_{2 \pi}^{\infty} \cos x d x∫2π∞​cosxdx

A torus is generated by rotating the circle x2+(y−6)2=25x^{2}+(y-6)^{2}=25x2+(y−6)2=25 about the x-axis. Find the volume enclosed by the torus. Round the answer to the nearest hundredth.

Find the volume obtained by rotating the region bounded by the given curves about y=−1y=-1y=−1 . y=sin⁡x,x=0,x=π,y=0y=\sin x, x=0, x=\pi, y=0y=sinx,x=0,x=π,y=0

Find the integral. ∫tan⁡2xsec⁡6xdx\int \tan ^{2} x \sec ^{6} x d x∫tan2xsec6xdx

Find the volume of the resulting solid if the region under the curve y=1(x2+3x+2)y=\frac{1}{\left(x^{2}+3 x+2\right)}y=(x2+3x+2)1​ from x=0x=0x=0 to x=1x=1x=1 is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. ∫0π/251−cos⁡4θdθ\int_{0}^{\pi / 2} 5 \sqrt{1-\cos 4 \theta} d \theta∫0π/2​51−cos4θ​dθ

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the integral using an appropriate trigonometric substitution. $\int \frac{x}{\sqrt{4-x^{2}}} d x$

Evaluate the indefinite integral. $\int x \cos 9 x d x$

Find a bound on the error in approximating the integral $\int_{1}^{9} \ln x^{9}$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n=10$ subintervals.

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-27}^{8} \frac{1}{\sqrt[3]{x}} d x$

Evaluate the integral. $\int_{x / 3}^{x / 2} 2 \cot ^{2} x d x$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{2} \frac{d x}{\sqrt{x^{3}+4}} ; \quad n=6$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{-\infty}^{\infty} \frac{3 e^{x}}{3+e^{2 x}} d x$

The region under the curve $y=2 \sin ^{2} x$ , $0 \leq x \leq \pi$ is rotated about the x-axis. Find the volume of the resulting solid.

Find the integral. $\int \frac{3 x-3}{x^{2}-x-2} d x$

Find the area bounded by the curves $y=3 \cos x$ and $y=3 \cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Find the area bounded by the curves $y=\cos x$ and $y=\cos ^{2} x$ between $x=0$ and $x=\frac{\pi}{2}$ .

Determine whether the integral converges or diverges. If it converges, find its value. $\int_{1}^{\infty} \frac{d x}{x^{5} \ln x}$

Evaluate the integral. $\int \frac{x^{2}}{\left(16-x^{2}\right)^{3 / 2}} d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{2 \pi}^{\infty} \cos x d x$

A torus is generated by rotating the circle $x^{2}+(y-6)^{2}=25$ about the x-axis. Find the volume enclosed by the torus. Round the answer to the nearest hundredth.

Find the volume obtained by rotating the region bounded by the given curves about $y=-1$ . $y=\sin x, x=0, x=\pi, y=0$

Find the integral. $\int \tan ^{2} x \sec ^{6} x d x$

Find the volume of the resulting solid if the region under the curve $y=\frac{1}{\left(x^{2}+3 x+2\right)}$ from $x=0$ to $x=1$ is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. $\int_{0}^{\pi / 2} 5 \sqrt{1-\cos 4 \theta} d \theta$