Question 1

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{2+\sin ^{2} x} d x ; \quad n=6$

Accepted Answer

A)  2.2028 
B)  2.7519 
C)  3.0265 
D)  2.4774 
A)  2.2028 
B)  2.7519 
C)  3.0265 
D)  2.4774

Question 2

For what values of K is the following integral improper? $\int_{0}^{x} \frac{5 x}{x^{2}-19 x+90} d x$

Accepted Answer

The answer of For what values of K is the...

Question 3

Evaluate the integral using an appropriate trigonometric substitution. $\int_{1}^{\sqrt{2}} \frac{1}{\left(2+x^{2}\right)^{3 / 2}} d x$

Accepted Answer

The answer of Evaluate the integral using an appropriate trigonometric...

Question 4

Evaluate the integral using the indicated trigonometric substitution. $\int \frac{d x}{x^{2} \sqrt{x^{2}-36}} ; x=6 \sec \theta$

Accepted Answer

The answer of Evaluate the integral using the indicated trigonometric...

Question 5

Evaluate the integral. $\int_{0}^{1} \frac{3}{x^{03}} d x$

Accepted Answer

The answer of Evaluate the integral. $\int_{0}^{1} \frac{3}{x^{03}} d x$...

Question 6

Find the integral. $\int \cos ^{5} x \sin ^{2} x d x$

Accepted Answer

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

Question 7

Find the integral. $\int x^{5} \ln x d x$

Accepted Answer

A)  $\frac{1}{6} x^{6}(6 \ln x-1)+C$ 
B)  $\frac{1}{36} x^{6}(\ln x-1)+C$ 
C)  $\frac{1}{6} x^{5}+\frac{1}{x}+C$ 
D)  $\frac{1}{36} x^{6}(6 \ln x-1)+C$ 
A)  $\frac{1}{6} x^{6}(6 \ln x-1)+C$ 
B)  $\frac{1}{36} x^{6}(\ln x-1)+C$ 
C)  $\frac{1}{6} x^{5}+\frac{1}{x}+C$ 
D)  $\frac{1}{36} x^{6}(6 \ln x-1)+C$

Question 8

Evaluate the integral using integration by parts with the indicated choices of u and dv. $\int 6 \theta \cos \theta d \theta, u=6 \theta, d v=\cos \theta d \theta$

Accepted Answer

A)  $6 \sin \theta+6 \cos \theta+C$ 
B)  $6 \theta \cos \theta+6 \sin \theta+C$ 
C)  $6 \theta \sin \theta+6 \cos \theta+C$ 
D)  $6 \sin \theta-6 \cos \theta+C$ 
E)  $\text { None of these }$ 
A)  $6 \sin \theta+6 \cos \theta+C$ 
B)  $6 \theta \cos \theta+6 \sin \theta+C$ 
C)  $6 \theta \sin \theta+6 \cos \theta+C$ 
D)  $6 \sin \theta-6 \cos \theta+C$ 
E)  $\text { None of these }$

Question 9

Evaluate the integral. $\int_{x / 6}^{\pi / 3} \frac{3 \ln (\tan x)}{7 \sin x \cos x} d x$

Accepted Answer

A)  $-\frac{1}{5}(\ln 2)^{2}$ 
B)  $-\frac{1}{2}(\ln 3)^{2}$ 
C)  0 
D)  $\frac{1}{6}(\ln 3)^{2}$ 
E)  $\frac{1}{2}(\ln 3)^{2}$ 
A)  $-\frac{1}{5}(\ln 2)^{2}$ 
B)  $-\frac{1}{2}(\ln 3)^{2}$ 
C)  0 
D)  $\frac{1}{6}(\ln 3)^{2}$ 
E)  $\frac{1}{2}(\ln 3)^{2}$

Question 10

Evaluate the integral to six decimal places. $\int_{0}^{1} \frac{x^{3}}{\sqrt{25-x^{2}}} d x$

Accepted Answer

The answer of Evaluate the integral to six decimal places....

Question 11

Let a and b be real numbers. What integral must appear in place of the question mark "?" to make the following statement true? $\int_{-\infty}^{a} \frac{10}{x^{2}+9} d x+\int_{a}^{\infty} \frac{10}{x^{2}+9} d x=?+\int_{b}^{\infty} \frac{10}{x^{2}+9} d x$

Accepted Answer

A)  $\int_{-\infty}^{a} \frac{9}{x^{2}+10} d x$ 
B)  $\int_{0}^{-\infty} \frac{10}{x^{2}-9} d x$ 
C)  $\int_{-\infty}^{b} \frac{10}{x^{2}+9} d x$ 
D)  $\int_{0}^{-\infty} \frac{10}{x^{2}+9} d x$ 
E)  $\text { none of these }$ 
A)  $\int_{-\infty}^{a} \frac{9}{x^{2}+10} d x$ 
B)  $\int_{0}^{-\infty} \frac{10}{x^{2}-9} d x$ 
C)  $\int_{-\infty}^{b} \frac{10}{x^{2}+9} d x$ 
D)  $\int_{0}^{-\infty} \frac{10}{x^{2}+9} d x$ 
E)  $\text { none of these }$

Question 12

Evaluate the integral. $\int_{0}^{1} \frac{x}{x+8} d x$

Accepted Answer

A)  $1-9 \ln 8+\sqrt{8} \ln 16$ 
B)  $\frac{1}{8 \ln 9}-\ln \sqrt{8}$ 
C)  $1-8(\ln 9-\ln 8)$ 
D)  $9-8 \ln 9+\ln \sqrt{8}$ 
E)  $\text { None of these }$ 
A)  $1-9 \ln 8+\sqrt{8} \ln 16$ 
B)  $\frac{1}{8 \ln 9}-\ln \sqrt{8}$ 
C)  $1-8(\ln 9-\ln 8)$ 
D)  $9-8 \ln 9+\ln \sqrt{8}$ 
E)  $\text { None of these }$

Question 13

Find the integral. $\int \sin ^{3} x \cos ^{6} x d x$

Accepted Answer

A)  $-\frac{1}{7} \cos ^{7} x+\frac{1}{5} \cos ^{5} x+C$ 
B)  $\frac{1}{7} \cos ^{7} x-\frac{1}{5} \cos ^{5} x+C$ 
C)  $-\frac{1}{9} \cos ^{9} x+\frac{1}{7} \cos ^{7} x+C$ 
D)  $\frac{1}{9} \cos ^{9} x-\frac{1}{7} \cos ^{7} x+C$ 
A)  $-\frac{1}{7} \cos ^{7} x+\frac{1}{5} \cos ^{5} x+C$ 
B)  $\frac{1}{7} \cos ^{7} x-\frac{1}{5} \cos ^{5} x+C$ 
C)  $-\frac{1}{9} \cos ^{9} x+\frac{1}{7} \cos ^{7} x+C$ 
D)  $\frac{1}{9} \cos ^{9} x-\frac{1}{7} \cos ^{7} x+C$

Question 14

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with $n=12$ and the formula $R=\frac{60 D}{\int_{0}^{24} C(t) d t}$ to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, $\boldsymbol{C}(\boldsymbol{t}) $ is the concentration of the dye in the aorta, and R is measured in liters per minute. Round your answer to one decimal place.
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{t} & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 \
\hline \boldsymbol{C}(\boldsymbol{t}) & 0 & 0 & 2.6 & 5.9 & 9.7 & 7.9 & 4.6 & 3.5 & 2.2 & 0.8 & 0.2 & 0.1 & 0 \
\hline
\end{array}$

Accepted Answer

6.5 liters

Question 15

Use the Table of Integrals to evaluate the integral. $\int \frac{\sqrt{49 x^{2}-1}}{x^{2}} d x$

Accepted Answer

A)  $7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|+C$ 
B)  $-\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|14 x+2 \sqrt{49 x^{2}-1}\right|+C$ 
C)  $\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|$ 
D)  $\frac{\sqrt{7 x^{2}-1}}{x}+\ln \left|x+\sqrt{7 x^{2}-1}\right|+C$ 
E)  $\frac{\sqrt{49 x^{2}-1}}{x}+\ln \left|7 x+\sqrt{49 x^{2}}\right|+C$ 
A)  $7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|+C$ 
B)  $-\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|14 x+2 \sqrt{49 x^{2}-1}\right|+C$ 
C)  $\frac{\sqrt{49 x^{2}-1}}{x}+7 \ln \left|7 x+\sqrt{49 x^{2}-1}\right|$ 
D)  $\frac{\sqrt{7 x^{2}-1}}{x}+\ln \left|x+\sqrt{7 x^{2}-1}\right|+C$ 
E)  $\frac{\sqrt{49 x^{2}-1}}{x}+\ln \left|7 x+\sqrt{49 x^{2}}\right|+C$

Question 16

A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 9 ft, what percentage of the total capacity is being used? Round the answer to the nearest tenth.

Accepted Answer

The answer of A water storage tank has the shape...

Question 17

Evaluate the integral. $\int \frac{7 d x}{\left(x^{2}+2 x+2\right)^{2}}$

Accepted Answer

A)  $\frac{1}{2}\left(\tan ^{-1}(x+2)+\frac{7}{x^{2}+2}\right)+C$ 
B)  $\frac{1}{2}\left(\tan ^{-1}(x+7)+\frac{1}{x^{2}+2 x+2}\right)+C$ 
C)  $\frac{7}{2}\left(\tan ^{-1}(x+1)+\frac{x+1}{x^{2}+2 x+2}\right)+C$ 
D)  $\frac{1}{2}\left(\tan (x+1)+\frac{17}{x^{2}+2 x+2}\right)+C$ 
E)  $\frac{7}{2}\left(\tan (x+2)+\frac{x+1}{x^{2}+2 x+2}\right)+C$ 
A)  $\frac{1}{2}\left(\tan ^{-1}(x+2)+\frac{7}{x^{2}+2}\right)+C$ 
B)  $\frac{1}{2}\left(\tan ^{-1}(x+7)+\frac{1}{x^{2}+2 x+2}\right)+C$ 
C)  $\frac{7}{2}\left(\tan ^{-1}(x+1)+\frac{x+1}{x^{2}+2 x+2}\right)+C$ 
D)  $\frac{1}{2}\left(\tan (x+1)+\frac{17}{x^{2}+2 x+2}\right)+C$ 
E)  $\frac{7}{2}\left(\tan (x+2)+\frac{x+1}{x^{2}+2 x+2}\right)+C$

Question 18

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{1} \frac{d x}{2 x+4} ; \quad n=7$

Accepted Answer

A)  0.2029 
B)  0.1088 
C)  0.1163 
D)  0.2326 
A)  0.2029 
B)  0.1088 
C)  0.1163 
D)  0.2326

Question 19

Use a table of integrals to evaluate the integral. $\int x \sqrt{2+2 x} d x$

Accepted Answer

A)  $\frac{2 \sqrt{2}}{15}(3 x-2) \sqrt{x+1}+C$ 
B)  $\frac{2 \sqrt{2}}{15}(2 x-1)(x+1)^{3 / 2}+C$ 
C)  $\frac{2}{15}(3 x-2) \sqrt{x+1}+C$ 
D)  $\frac{2 \sqrt{2}}{15}(3 x-2)(x+1)^{3 / 2}+C$ 
A)  $\frac{2 \sqrt{2}}{15}(3 x-2) \sqrt{x+1}+C$ 
B)  $\frac{2 \sqrt{2}}{15}(2 x-1)(x+1)^{3 / 2}+C$ 
C)  $\frac{2}{15}(3 x-2) \sqrt{x+1}+C$ 
D)  $\frac{2 \sqrt{2}}{15}(3 x-2)(x+1)^{3 / 2}+C$

Question 20

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{3 \pi}^{\infty} \cos x d x$

Accepted Answer

A)  6 
B)  0 
C)  3 
D)  Diverges 
A)  6 
B)  0 
C)  3 
D)  Diverges

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{2+\sin ^{2} x} d x ; \quad n=6$

For what values of K is the following integral improper? $\int_{0}^{x} \frac{5 x}{x^{2}-19 x+90} d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int_{1}^{\sqrt{2}} \frac{1}{\left(2+x^{2}\right)^{3 / 2}} d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac{d x}{x^{2} \sqrt{x^{2}-36}} ; x=6 \sec \theta$

Evaluate the integral. $\int_{0}^{1} \frac{3}{x^{03}} d x$

Find the integral. $\int \cos ^{5} x \sin ^{2} x d x$

Find the integral. $\int x^{5} \ln x d x$

Evaluate the integral using integration by parts with the indicated choices of u and dv. $\int 6 \theta \cos \theta d \theta, u=6 \theta, d v=\cos \theta d \theta$

Evaluate the integral. $\int_{x / 6}^{\pi / 3} \frac{3 \ln (\tan x)}{7 \sin x \cos x} d x$

Evaluate the integral to six decimal places. $\int_{0}^{1} \frac{x^{3}}{\sqrt{25-x^{2}}} d x$

Let a and b be real numbers. What integral must appear in place of the question mark "?" to make the following statement true? $\int_{-\infty}^{a} \frac{10}{x^{2}+9} d x+\int_{a}^{\infty} \frac{10}{x^{2}+9} d x=?+\int_{b}^{\infty} \frac{10}{x^{2}+9} d x$

Evaluate the integral. $\int_{0}^{1} \frac{x}{x+8} d x$

Find the integral. $\int \sin ^{3} x \cos ^{6} x d x$

Use the Table of Integrals to evaluate the integral. $\int \frac{\sqrt{49 x^{2}-1}}{x^{2}} d x$

A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 9 ft, what percentage of the total capacity is being used? Round the answer to the nearest tenth.

Evaluate the integral. $\int \frac{7 d x}{\left(x^{2}+2 x+2\right)^{2}}$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{1} \frac{d x}{2 x+4} ; \quad n=7$

Use a table of integrals to evaluate the integral. $\int x \sqrt{2+2 x} d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{3 \pi}^{\infty} \cos x d x$

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

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. ∫0π/22+sin⁡2xdx;n=6\int_{0}^{\pi / 2} \sqrt{2+\sin ^{2} x} d x ; \quad n=6∫0π/2​2+sin2x​dx;n=6

For what values of K is the following integral improper? ∫0x5xx2−19x+90dx\int_{0}^{x} \frac{5 x}{x^{2}-19 x+90} d x∫0x​x2−19x+905x​dx

Evaluate the integral using an appropriate trigonometric substitution. ∫121(2+x2)3/2dx\int_{1}^{\sqrt{2}} \frac{1}{\left(2+x^{2}\right)^{3 / 2}} d x∫12​​(2+x2)3/21​dx

Evaluate the integral using the indicated trigonometric substitution. ∫dxx2x2−36;x=6sec⁡θ\int \frac{d x}{x^{2} \sqrt{x^{2}-36}} ; x=6 \sec \theta∫x2x2−36​dx​;x=6secθ

Evaluate the integral. ∫013x03dx\int_{0}^{1} \frac{3}{x^{03}} d x∫01​x033​dx

Find the integral. ∫cos⁡5xsin⁡2xdx\int \cos ^{5} x \sin ^{2} x d x∫cos5xsin2xdx

Find the integral. ∫x5ln⁡xdx\int x^{5} \ln x d x∫x5lnxdx

Evaluate the integral using integration by parts with the indicated choices of u and dv. ∫6θcos⁡θdθ,u=6θ,dv=cos⁡θdθ\int 6 \theta \cos \theta d \theta, u=6 \theta, d v=\cos \theta d \theta∫6θcosθdθ,u=6θ,dv=cosθdθ

Evaluate the integral. ∫x/6π/33ln⁡(tan⁡x)7sin⁡xcos⁡xdx\int_{x / 6}^{\pi / 3} \frac{3 \ln (\tan x)}{7 \sin x \cos x} d x∫x/6π/3​7sinxcosx3ln(tanx)​dx

Evaluate the integral to six decimal places. ∫01x325−x2dx\int_{0}^{1} \frac{x^{3}}{\sqrt{25-x^{2}}} d x∫01​25−x2​x3​dx

Evaluate the integral. ∫01xx+8dx\int_{0}^{1} \frac{x}{x+8} d x∫01​x+8x​dx

Find the integral. ∫sin⁡3xcos⁡6xdx\int \sin ^{3} x \cos ^{6} x d x∫sin3xcos6xdx

Use the Table of Integrals to evaluate the integral. ∫49x2−1x2dx\int \frac{\sqrt{49 x^{2}-1}}{x^{2}} d x∫x249x2−1​​dx

A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 9 ft, what percentage of the total capacity is being used? Round the answer to the nearest tenth.

Evaluate the integral. ∫7dx(x2+2x+2)2\int \frac{7 d x}{\left(x^{2}+2 x+2\right)^{2}}∫(x2+2x+2)27dx​

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. ∫01dx2x+4;n=7\int_{0}^{1} \frac{d x}{2 x+4} ; \quad n=7∫01​2x+4dx​;n=7

Use a table of integrals to evaluate the integral. ∫x2+2xdx\int x \sqrt{2+2 x} d x∫x2+2x​dx

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫3π∞cos⁡xdx\int_{3 \pi}^{\infty} \cos x d x∫3π∞​cosxdx

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

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{\pi / 2} \sqrt{2+\sin ^{2} x} d x ; \quad n=6$

For what values of K is the following integral improper? $\int_{0}^{x} \frac{5 x}{x^{2}-19 x+90} d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int_{1}^{\sqrt{2}} \frac{1}{\left(2+x^{2}\right)^{3 / 2}} d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac{d x}{x^{2} \sqrt{x^{2}-36}} ; x=6 \sec \theta$

Evaluate the integral. $\int_{0}^{1} \frac{3}{x^{03}} d x$

Find the integral. $\int \cos ^{5} x \sin ^{2} x d x$

Find the integral. $\int x^{5} \ln x d x$

Evaluate the integral using integration by parts with the indicated choices of u and dv. $\int 6 \theta \cos \theta d \theta, u=6 \theta, d v=\cos \theta d \theta$

Evaluate the integral. $\int_{x / 6}^{\pi / 3} \frac{3 \ln (\tan x)}{7 \sin x \cos x} d x$

Evaluate the integral to six decimal places. $\int_{0}^{1} \frac{x^{3}}{\sqrt{25-x^{2}}} d x$

Evaluate the integral. $\int_{0}^{1} \frac{x}{x+8} d x$

Find the integral. $\int \sin ^{3} x \cos ^{6} x d x$

Use the Table of Integrals to evaluate the integral. $\int \frac{\sqrt{49 x^{2}-1}}{x^{2}} d x$

Evaluate the integral. $\int \frac{7 d x}{\left(x^{2}+2 x+2\right)^{2}}$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int_{0}^{1} \frac{d x}{2 x+4} ; \quad n=7$

Use a table of integrals to evaluate the integral. $\int x \sqrt{2+2 x} d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int_{3 \pi}^{\infty} \cos x d x$