Question 1

Calculate the area between the curve $y=\sin ^{2} \theta \cos \theta$ and the x-axis between $\theta=0$ and $\theta=\pi / 6$ .Round your answer to 2 decimal places.

Accepted Answer

The answer of Calculate the area between the curve \(y=\sin...

Question 2

Evaluate exactly: $\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}$ .

Accepted Answer

A)  $\frac{693}{1000}$ 
B)  $\ln 2$ 
C)  $\ln \frac{1}{2}$ 
D)  $-\frac{6931}{10,000}$ 
A)  $\frac{693}{1000}$ 
B)  $\ln 2$ 
C)  $\ln \frac{1}{2}$ 
D)  $-\frac{6931}{10,000}$

Question 3

If $\int_{0}^{2} \frac{d x}{(x-1)^{6 / 5}}$ converges, find its value.Otherwise, enter "DNC".

Accepted Answer

The answer of If $\int_{0}^{2} \frac{d x}{(x-1)^{6 / 5}}$ converges,...

Question 4

Compute $\int_{0}^{3} x \sqrt{16+x^{2}} d x$ .Round to 3 decimal places.

Accepted Answer

The answer of Compute $\int_{0}^{3} x \sqrt{16+x^{2}} d x$ .Round...

Question 5

Evaluate exactly: $\int_{0}^{3} \frac{x^{2} d x}{x^{3}+4}$ .

Accepted Answer

The answer of Evaluate exactly: $\int_{0}^{3} \frac{x^{2} d x}{x^{3}+4}$ ....

Question 6

What is the value of A if $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}$ ?

Accepted Answer

A) 5 
B) 6 
C) 7 
D) None of the above. 
A) 5 
B) 6 
C) 7 
D) None of the above.

Question 7

$\int \frac{1}{\sqrt{4 x+8}} d x=(4 x+8)^{1 / 2}+C$ .

Accepted Answer

A) True 
 B)False

Question 8

$\int \frac{1}{6 x+4} d x=\frac{1}{6} \ln |6 x+4|+C$ .

Accepted Answer

A) True 
 B)False

Question 9

$\int \theta^{2} \cos (a \theta) d \theta=\frac{\theta^{2}}{a} \sin (a \theta)+\frac{2 \theta}{a^{2}} \cos (a \theta)+C$ , where a is a constant.

Accepted Answer

A) True 
 B)False

Question 10

Calculate $\int y \sec ^{2} y d y$ .

Accepted Answer

A)  $y \tan y+\ln |\cos y|+C$ 
B)  $y \tan y-\ln |\cos y|+C$ 
C)  $\frac{y^{2}}{2}-\tan y+C$ 
D)  $\frac{y^{2}}{2} \tan y+C$ 
A)  $y \tan y+\ln |\cos y|+C$ 
B)  $y \tan y-\ln |\cos y|+C$ 
C)  $\frac{y^{2}}{2}-\tan y+C$ 
D)  $\frac{y^{2}}{2} \tan y+C$

Question 11

$\int_{2}^{4} \frac{e^{2 x}}{\sqrt{e^{x}-1}} d x=\frac{2}{3}\left(e^{4}-1\right)^{\frac{3}{2}}+2\left(e^{4}-1\right)^{\frac{1}{2}}-\frac{2}{3}\left(e^{2}-1\right)^{\frac{3}{2}}-2\left(e^{2}-1\right)^{\frac{1}{2}}$ .

Accepted Answer

A) True 
 B)False

Question 12

Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of $r(t)=18 e^{-0.1 t}$ psi per second at time t in seconds.By how many total psi has the pressure decreased during the first minute? Round to 2 decimal places.

Accepted Answer

The answer of Fuel pressure in the fuel tanks of...

Question 13

Evaluate $\int_{-7}^{18} \arctan (x) d x$ "symbolically" (plug in the limits but don't evaluate).Hint: Integrate by parts using $\frac{d}{d x} \arctan x=\frac{1}{1+x^{2}}$ .

Accepted Answer

A)  $18 \arctan 18-7 \arctan (-7)+\ln \left(1+18^{2}\right)+\ln \left(1+7^{2}\right)$ 
B)  $18 \arctan 18+7 \arctan (-7)-\ln \left(1+18^{2}\right)-\ln \left(1+7^{2}\right)$ 
C)  $18 \arctan 18-7 \arctan (-7)+\frac{1}{2} \ln \left(1+18^{2}\right)-\frac{1}{2} \ln \left(1+7^{2}\right)$ 
D)  $18 \arctan 18+7 \arctan (-7)-\frac{1}{2} \ln \left(1+18^{2}\right)+\frac{1}{2} \ln \left(1+7^{2}\right)$ 
A)  $18 \arctan 18-7 \arctan (-7)+\ln \left(1+18^{2}\right)+\ln \left(1+7^{2}\right)$ 
B)  $18 \arctan 18+7 \arctan (-7)-\ln \left(1+18^{2}\right)-\ln \left(1+7^{2}\right)$ 
C)  $18 \arctan 18-7 \arctan (-7)+\frac{1}{2} \ln \left(1+18^{2}\right)-\frac{1}{2} \ln \left(1+7^{2}\right)$ 
D)  $18 \arctan 18+7 \arctan (-7)-\frac{1}{2} \ln \left(1+18^{2}\right)+\frac{1}{2} \ln \left(1+7^{2}\right)$

Question 14

The actual value of $\int_{0}^{10} 5 \ln \left(x^{2}+1\right) d x$ lies:

Accepted Answer

A) between 142.5 and 145. 
B) between 145 and 147.5. 
C) between 147.5 and 150. 
D) between 150 and 152.5. 
A) between 142.5 and 145. 
B) between 145 and 147.5. 
C) between 147.5 and 150. 
D) between 150 and 152.5.

Question 15

Evaluate $\int 6 x^{7} d x$ .

Accepted Answer

A)  $\frac{3}{4} x^{8}+C$ 
B)  $x^{8}+C$ 
C)  $\frac{6}{7} x^{8}+C$ 
D)  $\frac{7}{8} x^{8}+C$ 
A)  $\frac{3}{4} x^{8}+C$ 
B)  $x^{8}+C$ 
C)  $\frac{6}{7} x^{8}+C$ 
D)  $\frac{7}{8} x^{8}+C$

Question 16

$\int \frac{4 x^{3}+\cos x}{x^{4}+\sin x} d x=\ln \left|x^{4}+\sin x\right|+C$ .

Accepted Answer

A) True 
 B)False

Question 17

Integrate $\int(\tan 7 x+\cos 7 x) d x$ .

Accepted Answer

A)  $\frac{\ln |\cos 7 x|}{7}-\frac{\sin 7 x}{7}+C$ 
B)  $-\frac{\ln |\cos 7 x|}{7}+\frac{\sin 7 x}{7}+C$ 
C)  $\frac{\cot 7 x}{7}-\frac{\sin 7 x}{7}+C$ 
D)  $-\frac{\cot 7 x}{7}+\frac{\sin 7 x}{7}+C$ 
A)  $\frac{\ln |\cos 7 x|}{7}-\frac{\sin 7 x}{7}+C$ 
B)  $-\frac{\ln |\cos 7 x|}{7}+\frac{\sin 7 x}{7}+C$ 
C)  $\frac{\cot 7 x}{7}-\frac{\sin 7 x}{7}+C$ 
D)  $-\frac{\cot 7 x}{7}+\frac{\sin 7 x}{7}+C$

Question 18

$\int \frac{5-\ln x}{x} d x=5 \ln |x|-\frac{(\ln |x|)^{2}}{2}+C$ .

Accepted Answer

A) True 
 B)False

Question 19

For $f(x)=x^{2} e^{10 x}$ , find a function $F(x)$ such that $F^{\prime}(x)=f(x)$ and $F(0)=0$ .

Accepted Answer

A)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)-\frac{1}{500}$ 
B)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)$ 
C)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x\right)$ 
D)  $e^{10 x}\left(\frac{x^{3}}{30}\right)$ 
A)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)-\frac{1}{500}$ 
B)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)$ 
C)  $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x\right)$ 
D)  $e^{10 x}\left(\frac{x^{3}}{30}\right)$

Question 20

Find the exact value of $\int_{1}^{3} x \sqrt[3]{x-1} d x$ .

Accepted Answer

A)  $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}+\frac{9}{28}$ 
B)  $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}-\frac{33}{28}$ 
C)  $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}$ 
D)  $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}$ 
A)  $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}+\frac{9}{28}$ 
B)  $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}-\frac{33}{28}$ 
C)  $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}$ 
D)  $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}$

Exam 7: Integration

Calculate the area between the curve y=sin⁡2θcos⁡θy=\sin ^{2} \theta \cos \thetay=sin2θcosθ and the x-axis between θ=0\theta=0θ=0 and θ=π/6\theta=\pi / 6θ=π/6 .Round your answer to 2 decimal places.

Evaluate exactly: ∫x/6x/2dθtan⁡θ\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}∫x/6x/2​tanθdθ​ .

If ∫02dx(x−1)6/5\int_{0}^{2} \frac{d x}{(x-1)^{6 / 5}}∫02​(x−1)6/5dx​ converges, find its value.Otherwise, enter "DNC".

Compute ∫03x16+x2dx\int_{0}^{3} x \sqrt{16+x^{2}} d x∫03​x16+x2​dx .Round to 3 decimal places.

Evaluate exactly: ∫03x2dxx3+4\int_{0}^{3} \frac{x^{2} d x}{x^{3}+4}∫03​x3+4x2dx​ .

What is the value of A if ∫0ln⁡6xexdx=−ln⁡66+A6\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}∫0ln6​exx​dx=−6ln6​+6A​ ?

∫14x+8dx=(4x+8)1/2+C\int \frac{1}{\sqrt{4 x+8}} d x=(4 x+8)^{1 / 2}+C∫4x+8​1​dx=(4x+8)1/2+C .

∫16x+4dx=16ln⁡∣6x+4∣+C\int \frac{1}{6 x+4} d x=\frac{1}{6} \ln |6 x+4|+C∫6x+41​dx=61​ln∣6x+4∣+C .

∫θ2cos⁡(aθ)dθ=θ2asin⁡(aθ)+2θa2cos⁡(aθ)+C\int \theta^{2} \cos (a \theta) d \theta=\frac{\theta^{2}}{a} \sin (a \theta)+\frac{2 \theta}{a^{2}} \cos (a \theta)+C∫θ2cos(aθ)dθ=aθ2​sin(aθ)+a22θ​cos(aθ)+C , where a is a constant.

Calculate ∫ysec⁡2ydy\int y \sec ^{2} y d y∫ysec2ydy .

Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of r(t)=18e−0.1tr(t)=18 e^{-0.1 t}r(t)=18e−0.1t psi per second at time t in seconds.By how many total psi has the pressure decreased during the first minute? Round to 2 decimal places.

Evaluate ∫−718arctan⁡(x)dx\int_{-7}^{18} \arctan (x) d x∫−718​arctan(x)dx "symbolically" (plug in the limits but don't evaluate).Hint: Integrate by parts using ddxarctan⁡x=11+x2\frac{d}{d x} \arctan x=\frac{1}{1+x^{2}}dxd​arctanx=1+x21​ .

The actual value of ∫0105ln⁡(x2+1)dx\int_{0}^{10} 5 \ln \left(x^{2}+1\right) d x∫010​5ln(x2+1)dx lies:

Evaluate ∫6x7dx\int 6 x^{7} d x∫6x7dx .

∫4x3+cos⁡xx4+sin⁡xdx=ln⁡∣x4+sin⁡x∣+C\int \frac{4 x^{3}+\cos x}{x^{4}+\sin x} d x=\ln \left|x^{4}+\sin x\right|+C∫x4+sinx4x3+cosx​dx=ln​x4+sinx​+C .

Integrate ∫(tan⁡7x+cos⁡7x)dx\int(\tan 7 x+\cos 7 x) d x∫(tan7x+cos7x)dx .

∫5−ln⁡xxdx=5ln⁡∣x∣−(ln⁡∣x∣)22+C\int \frac{5-\ln x}{x} d x=5 \ln |x|-\frac{(\ln |x|)^{2}}{2}+C∫x5−lnx​dx=5ln∣x∣−2(ln∣x∣)2​+C .

For f(x)=x2e10xf(x)=x^{2} e^{10 x}f(x)=x2e10x , find a function F(x)F(x)F(x) such that F′(x)=f(x)F^{\prime}(x)=f(x)F′(x)=f(x) and F(0)=0F(0)=0F(0)=0 .

Find the exact value of ∫13xx−13dx\int_{1}^{3} x \sqrt[3]{x-1} d x∫13​x3x−1​dx .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Calculate the area between the curve $y=\sin ^{2} \theta \cos \theta$ and the x-axis between $\theta=0$ and $\theta=\pi / 6$ .Round your answer to 2 decimal places.

Evaluate exactly: $\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}$ .

If $\int_{0}^{2} \frac{d x}{(x-1)^{6 / 5}}$ converges, find its value.Otherwise, enter "DNC".

Compute $\int_{0}^{3} x \sqrt{16+x^{2}} d x$ .Round to 3 decimal places.

Evaluate exactly: $\int_{0}^{3} \frac{x^{2} d x}{x^{3}+4}$ .

What is the value of A if $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}$ ?

$\int \frac{1}{\sqrt{4 x+8}} d x=(4 x+8)^{1 / 2}+C$ .

$\int \frac{1}{6 x+4} d x=\frac{1}{6} \ln |6 x+4|+C$ .

$\int \theta^{2} \cos (a \theta) d \theta=\frac{\theta^{2}}{a} \sin (a \theta)+\frac{2 \theta}{a^{2}} \cos (a \theta)+C$ , where a is a constant.

Calculate $\int y \sec ^{2} y d y$ .

Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of $r(t)=18 e^{-0.1 t}$ psi per second at time t in seconds.By how many total psi has the pressure decreased during the first minute? Round to 2 decimal places.

Evaluate $\int_{-7}^{18} \arctan (x) d x$ "symbolically" (plug in the limits but don't evaluate).Hint: Integrate by parts using $\frac{d}{d x} \arctan x=\frac{1}{1+x^{2}}$ .

The actual value of $\int_{0}^{10} 5 \ln \left(x^{2}+1\right) d x$ lies:

Evaluate $\int 6 x^{7} d x$ .

$\int \frac{4 x^{3}+\cos x}{x^{4}+\sin x} d x=\ln \left|x^{4}+\sin x\right|+C$ .

Integrate $\int(\tan 7 x+\cos 7 x) d x$ .

$\int \frac{5-\ln x}{x} d x=5 \ln |x|-\frac{(\ln |x|)^{2}}{2}+C$ .

For $f(x)=x^{2} e^{10 x}$ , find a function $F(x)$ such that $F^{\prime}(x)=f(x)$ and $F(0)=0$ .

Find the exact value of $\int_{1}^{3} x \sqrt[3]{x-1} d x$ .