Question 1

Evaluate the integral. $\int_{0}^{1 / 2} 12 x \cos \pi x d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{0}^{1 / 2} 12...

Question 2

Household electricity is supplied in the form of alternating current that varies from 170 V to -170 V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the function E(t), where t is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of $[E(t)]^{2}$ over one cycle. Calculate the RMS voltage of household current. Round your answer to the nearest integer. $E(t)=230 \sin (120 \pi t)$

Accepted Answer

The answer of Household electricity is supplied in the form...

Question 3

Find the integral. $\int \frac{5 x^{2}-9 x+6}{x^{3}-2 x^{2}+x} d x$

Accepted Answer

A)  $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{1}{x-1}+C$ 
B)  $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{1}{x-1}+C$ 
C)  $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{2}{x-1}+C$ 
D)  $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{2}{x-1}+C$ 
A)  $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{1}{x-1}+C$ 
B)  $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{1}{x-1}+C$ 
C)  $\ln \left|\frac{x-1}{x^{6}}\right|-\frac{2}{x-1}+C$ 
D)  $\ln \left|\frac{x^{6}}{x-1}\right|-\frac{2}{x-1}+C$

Question 4

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation. $2 \int_{2}^{3} e^{-\sqrt{x}} d x, \quad n=6$

Accepted Answer

A)  $-0.00008$ 
B)  $0.60004$ 
C)  0.00008 
D)  1.00008 
E)  $-0.00096$ 
A)  $-0.00008$ 
B)  $0.60004$ 
C)  0.00008 
D)  1.00008 
E)  $-0.00096$

Question 5

Evaluate the following integral. $\int_{0}^{1} \frac{3 \ln 2 x}{\sqrt{x}} d x$

Accepted Answer

The answer of Evaluate the following integral. \(\int_{0}^{1} \frac{3 \ln...

Question 6

Use long division to evaluate the integral. $\int_{0}^{1} \frac{x^{3}+4 x^{2}-12 x+1}{x^{2}+4 x-12} d x$ The choices are rounded to 3 decimal places.

Accepted Answer

A)  $-4.606$ 
B)  $5.394$ 
C)  $0.394$ 
D)  $-9.606$ 
E)  $10.394$ 
A)  $-4.606$ 
B)  $5.394$ 
C)  $0.394$ 
D)  $-9.606$ 
E)  $10.394$

Question 7

Use the Table of Integrals to evaluate the integral. $\int e^{4 x} \sin 2 x d x$

Accepted Answer

A)  $-\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$ 
B)  $\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$ 
C)  $\frac{1}{5} e^{4 x} \sin 2 x-\frac{3}{10} e^{4 x} \cos 2 x+C$ 
D)  $-\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$ 
E)  $\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$ 
A)  $-\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$ 
B)  $\frac{1}{5} e^{4 x} \sin 2 x+\frac{1}{10} e^{4 x} \cos 2 x+C$ 
C)  $\frac{1}{5} e^{4 x} \sin 2 x-\frac{3}{10} e^{4 x} \cos 2 x+C$ 
D)  $-\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$ 
E)  $\frac{1}{5} e^{4 x} \sin 2 x-\frac{1}{10} e^{4 x} \cos 2 x+C$

Question 8

The region under the graph of $y=\frac{7}{x(x+1)}$ on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Accepted Answer

The answer of The region under the graph of $y=\frac{7}{x(x+1)}$...

Question 9

Evaluate the integral. $\int 8\left(\frac{x-1}{x^{2}+2 x}\right) d x$

Accepted Answer

A)  $8\left(\frac{1}{3} x-\frac{1}{3} \ln x\right)+C$ 
B)  $-8(\ln x-x)+C$ 
C)  $8\left(\frac{3}{2} \ln (x+2)-\frac{1}{2} \ln x\right)+C$ 
D)  $-8\left(\frac{\sqrt{2}}{2} \arctan x\right)+C$ 
E)  $\text { None of these }$ 
A)  $8\left(\frac{1}{3} x-\frac{1}{3} \ln x\right)+C$ 
B)  $-8(\ln x-x)+C$ 
C)  $8\left(\frac{3}{2} \ln (x+2)-\frac{1}{2} \ln x\right)+C$ 
D)  $-8\left(\frac{\sqrt{2}}{2} \arctan x\right)+C$ 
E)  $\text { None of these }$

Question 10

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x^{2} \ln x}$

Accepted Answer

A)  $-\frac{1}{4}$ 
B)  $\frac{1}{4}$ 
C)  $\text { divergent }$ 
D)  2 
E)  $\frac{1}{2}$ 
A)  $-\frac{1}{4}$ 
B)  $\frac{1}{4}$ 
C)  $\text { divergent }$ 
D)  2 
E)  $\frac{1}{2}$

Question 11

Evaluate the integral. $\int \frac{1+8 e^{x}}{1-e^{x}} d x$

Accepted Answer

The answer of Evaluate the integral. \(\int \frac{1+8 e^{x}}{1-e^{x}} d...

Question 12

Find the area of the region bounded by the hyperbola $9 x^{2}-4 y^{2}=36$ and the line $x=6$ .

Accepted Answer

The answer of Find the area of the region bounded...

Question 13

Evaluate the integral. $\int_{x / 6}^{x / 3} \frac{x}{\sin ^{2} x} d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{x / 6}^{x /...

Question 14

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int_{4}^{25} \frac{\sqrt{x}}{x-100} d x$ Round the answer to four decimal places.

Accepted Answer

The answer of Make a substitution to express the integrand...

Question 15

Use the Trapezoidal Rule to approximate $\int_{2}^{3} e^{3 / x} d x$ for $n=4$ . Round the result to four decimal places.

Accepted Answer

The answer of Use the Trapezoidal Rule to approximate \(\int_{2}^{3}...

Question 16

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt{9-x^{2}} d x$

Accepted Answer

A)  $\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$ 
B)  $-\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$ 
C)  $\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$ 
D)  $-\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$ 
A)  $\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$ 
B)  $-\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$ 
C)  $\frac{1}{3} x^{2}\left(9-x^{2}\right)^{3 / 2}+C$ 
D)  $-\frac{1}{3}\left(9-x^{2}\right)^{3 / 2}+C$

Question 17

Find the integral using an appropriate trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+36}} d x$

Accepted Answer

A)  $\frac{1}{3}\left(x^{2}-36\right)^{3 / 2} \sqrt{x^{2}+36}+C$ 
B)  $\frac{1}{3}\left(x^{2}+72\right) \sqrt{x^{2}+36}+C$ 
C)  $\frac{1}{3}\left(x^{2}-72\right) \sqrt{x^{2}+36}+C$ 
D)  $\frac{1}{3}\left(x^{2}+36\right)^{3 / 2} \sqrt{x^{2}+36}+C$ 
A)  $\frac{1}{3}\left(x^{2}-36\right)^{3 / 2} \sqrt{x^{2}+36}+C$ 
B)  $\frac{1}{3}\left(x^{2}+72\right) \sqrt{x^{2}+36}+C$ 
C)  $\frac{1}{3}\left(x^{2}-72\right) \sqrt{x^{2}+36}+C$ 
D)  $\frac{1}{3}\left(x^{2}+36\right)^{3 / 2} \sqrt{x^{2}+36}+C$

Question 18

Evaluate the integral. $2 \int_{0}^{\pi / 2} \sin ^{3} \theta \cos ^{2} \theta d \theta$

Accepted Answer

The answer of Evaluate the integral. \(2 \int_{0}^{\pi / 2}...

Question 19

Evaluate the integral. $ \int_{0}^{1} \frac{3 x^{3}+9 x^{2}+13 x+5}{(x+1)^{2}\left(x^{2}+1\right)} d x $

Accepted Answer

The answer of Evaluate the integral. \( \int_{0}^{1} \frac{3 x^{3}+9...

Question 20

Evaluate the integral. $\int x \sin 5 x \cos 5 x d x$

Accepted Answer

A)  $-\frac{1}{12} x \sin ^{2} x+\frac{1}{12} \cos x+C$ 
B)  $\frac{1}{288}[\sin 288 x-288 \cos 288 x]+C$ 
C)  $-\frac{1}{12} \cos x \sin x+\frac{1}{12} x+C$ 
D)  $\frac{1}{288}[\sin 12 x-12 \cos 12 x]+C$ 
E)  $\text { none of these }$ 
A)  $-\frac{1}{12} x \sin ^{2} x+\frac{1}{12} \cos x+C$ 
B)  $\frac{1}{288}[\sin 288 x-288 \cos 288 x]+C$ 
C)  $-\frac{1}{12} \cos x \sin x+\frac{1}{12} x+C$ 
D)  $\frac{1}{288}[\sin 12 x-12 \cos 12 x]+C$ 
E)  $\text { none of these }$

Evaluate the integral. $\int_{0}^{1 / 2} 12 x \cos \pi x d x$

Find the integral. $\int \frac{5 x^{2}-9 x+6}{x^{3}-2 x^{2}+x} d x$

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation. $2 \int_{2}^{3} e^{-\sqrt{x}} d x, \quad n=6$

Evaluate the following integral. $\int_{0}^{1} \frac{3 \ln 2 x}{\sqrt{x}} d x$

Use long division to evaluate the integral. $\int_{0}^{1} \frac{x^{3}+4 x^{2}-12 x+1}{x^{2}+4 x-12} d x$ The choices are rounded to 3 decimal places.

Use the Table of Integrals to evaluate the integral. $\int e^{4 x} \sin 2 x d x$

The region under the graph of $y=\frac{7}{x(x+1)}$ on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Evaluate the integral. $\int 8\left(\frac{x-1}{x^{2}+2 x}\right) d x$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x^{2} \ln x}$

Evaluate the integral. $\int \frac{1+8 e^{x}}{1-e^{x}} d x$

Find the area of the region bounded by the hyperbola $9 x^{2}-4 y^{2}=36$ and the line $x=6$ .

Evaluate the integral. $\int_{x / 6}^{x / 3} \frac{x}{\sin ^{2} x} d x$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int_{4}^{25} \frac{\sqrt{x}}{x-100} d x$ Round the answer to four decimal places.

Use the Trapezoidal Rule to approximate $\int_{2}^{3} e^{3 / x} d x$ for $n=4$ . Round the result to four decimal places.

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt{9-x^{2}} d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+36}} d x$

Evaluate the integral. $2 \int_{0}^{\pi / 2} \sin ^{3} \theta \cos ^{2} \theta d \theta$

Evaluate the integral. $\int_{0}^{1} \frac{3 x^{3}+9 x^{2}+13 x+5}{(x+1)^{2}\left(x^{2}+1\right)} d x$

Evaluate the integral. $\int x \sin 5 x \cos 5 x d x$

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

Evaluate the integral. ∫01/212xcos⁡πxdx\int_{0}^{1 / 2} 12 x \cos \pi x d x∫01/2​12xcosπxdx

Find the integral. ∫5x2−9x+6x3−2x2+xdx\int \frac{5 x^{2}-9 x+6}{x^{3}-2 x^{2}+x} d x∫x3−2x2+x5x2−9x+6​dx

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation. 2∫23e−xdx,n=62 \int_{2}^{3} e^{-\sqrt{x}} d x, \quad n=62∫23​e−x​dx,n=6

Evaluate the following integral. ∫013ln⁡2xxdx\int_{0}^{1} \frac{3 \ln 2 x}{\sqrt{x}} d x∫01​x​3ln2x​dx

Use long division to evaluate the integral. ∫01x3+4x2−12x+1x2+4x−12dx\int_{0}^{1} \frac{x^{3}+4 x^{2}-12 x+1}{x^{2}+4 x-12} d x∫01​x2+4x−12x3+4x2−12x+1​dx The choices are rounded to 3 decimal places.

Use the Table of Integrals to evaluate the integral. ∫e4xsin⁡2xdx\int e^{4 x} \sin 2 x d x∫e4xsin2xdx

The region under the graph of y=7x(x+1)y=\frac{7}{x(x+1)}y=x(x+1)7​ on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Evaluate the integral. ∫8(x−1x2+2x)dx\int 8\left(\frac{x-1}{x^{2}+2 x}\right) d x∫8(x2+2xx−1​)dx

Evaluate the integral. ∫1∞dxx2ln⁡x\int_{1}^{\infty} \frac{d x}{x^{2} \ln x}∫1∞​x2lnxdx​

Evaluate the integral. ∫1+8ex1−exdx\int \frac{1+8 e^{x}}{1-e^{x}} d x∫1−ex1+8ex​dx

Find the area of the region bounded by the hyperbola 9x2−4y2=369 x^{2}-4 y^{2}=369x2−4y2=36 and the line x=6x=6x=6 .

Evaluate the integral. ∫x/6x/3xsin⁡2xdx\int_{x / 6}^{x / 3} \frac{x}{\sin ^{2} x} d x∫x/6x/3​sin2xx​dx

Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫425xx−100dx\int_{4}^{25} \frac{\sqrt{x}}{x-100} d x∫425​x−100x​​dx Round the answer to four decimal places.

Use the Trapezoidal Rule to approximate ∫23e3/xdx\int_{2}^{3} e^{3 / x} d x∫23​e3/xdx for n=4n=4n=4 . Round the result to four decimal places.

Find the integral using an appropriate trigonometric substitution. ∫x9−x2dx\int x \sqrt{9-x^{2}} d x∫x9−x2​dx

Find the integral using an appropriate trigonometric substitution. ∫x3x2+36dx\int \frac{x^{3}}{\sqrt{x^{2}+36}} d x∫x2+36​x3​dx

Evaluate the integral. 2∫0π/2sin⁡3θcos⁡2θdθ2 \int_{0}^{\pi / 2} \sin ^{3} \theta \cos ^{2} \theta d \theta2∫0π/2​sin3θcos2θdθ

Evaluate the integral. ∫013x3+9x2+13x+5(x+1)2(x2+1)dx \int_{0}^{1} \frac{3 x^{3}+9 x^{2}+13 x+5}{(x+1)^{2}\left(x^{2}+1\right)} d x ∫01​(x+1)2(x2+1)3x3+9x2+13x+5​dx

Evaluate the integral. ∫xsin⁡5xcos⁡5xdx\int x \sin 5 x \cos 5 x d x∫xsin5xcos5xdx

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

Evaluate the integral. $\int_{0}^{1 / 2} 12 x \cos \pi x d x$

Find the integral. $\int \frac{5 x^{2}-9 x+6}{x^{3}-2 x^{2}+x} d x$

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation. $2 \int_{2}^{3} e^{-\sqrt{x}} d x, \quad n=6$

Evaluate the following integral. $\int_{0}^{1} \frac{3 \ln 2 x}{\sqrt{x}} d x$

Use long division to evaluate the integral. $\int_{0}^{1} \frac{x^{3}+4 x^{2}-12 x+1}{x^{2}+4 x-12} d x$ The choices are rounded to 3 decimal places.

Use the Table of Integrals to evaluate the integral. $\int e^{4 x} \sin 2 x d x$

The region under the graph of $y=\frac{7}{x(x+1)}$ on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Evaluate the integral. $\int 8\left(\frac{x-1}{x^{2}+2 x}\right) d x$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x^{2} \ln x}$

Evaluate the integral. $\int \frac{1+8 e^{x}}{1-e^{x}} d x$

Find the area of the region bounded by the hyperbola $9 x^{2}-4 y^{2}=36$ and the line $x=6$ .

Evaluate the integral. $\int_{x / 6}^{x / 3} \frac{x}{\sin ^{2} x} d x$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int_{4}^{25} \frac{\sqrt{x}}{x-100} d x$ Round the answer to four decimal places.

Use the Trapezoidal Rule to approximate $\int_{2}^{3} e^{3 / x} d x$ for $n=4$ . Round the result to four decimal places.

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt{9-x^{2}} d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac{x^{3}}{\sqrt{x^{2}+36}} d x$

Evaluate the integral. $2 \int_{0}^{\pi / 2} \sin ^{3} \theta \cos ^{2} \theta d \theta$

Evaluate the integral. $\int_{0}^{1} \frac{3 x^{3}+9 x^{2}+13 x+5}{(x+1)^{2}\left(x^{2}+1\right)} d x$

Evaluate the integral. $\int x \sin 5 x \cos 5 x d x$