7.4.3Can I use integration by parts again?

Integration by Parts with Substitution

7-199.

Review your summary of the integration by parts method, from problem 7-191. Then, with your team, evaluate the integrals below.

$\int x \sin(x)dx$

$\int x(3x - 5)^{100}dx$

7-200.

PLAY IT AGAIN, SAM

Samantha tried to use integration by parts for the integral $\int x ^ { 2 } e ^ { x } d x$ . She started with $f = x^2$ and $dg = e^xdx$ . Follow her work below.

$f=x^2$
$\downarrow$
$df=2xdx$

$dg=e^xdx$
$\downarrow$
$g=e^x$

Therefore,

$\int_{ }^{ }x^2e^xdx=$ $x^2e^x-2\int_{ }^{ }xe^xdx$

Samantha’s new integral appears nicer than the original. However, it still contains a product. What should Samantha do next?
Finish her work to evaluate $\int x ^ { 2 } e ^ { x } d x$ .
Evaluating $\int x ^ { 2 } e ^ { x } d x$ required using the integration by parts process twice. How many times does the integration by parts process need to be used to evaluate $\int x ^ { 5 } e ^ { x } d x$ ? Explain your thinking.
Similarly, evaluate $\int x^2 \sin(x)dx$ .

7-201.

REAPPEARING ACT

Elise is trying to evaluate $\int e^x \sin(x)dx$ using integration by parts and is getting frustrated. No matter what she assigns to $f$ and $dg$ , none of the terms seem to disappear. The new integral always contains $e^x$ and either $\sin(x)$ or $\cos(x)$ . Her teammate, Edward, notices that when she uses the process twice, the new integral looks familiar. Edward thinks this might be a step in the right direction. Examine their work:

$f\left(x\right)=e^x$
$\downarrow$
$df=e^xdx$

$dg=\sin\left(x\right)dx$
$\downarrow$
$g\left(x\right)=-\cos\left(x\right)$

Therefore:
$\int_{ }^{ }e^x\sin(x)dx=-e^x\cos(x)+ \underbrace{\int_{ }^{ }e^x\cos(x)dx}$

Using the process again:

$\quad \ \nearrow \\ \swarrow$

$f\left(x\right)=e^x$
$\downarrow$
$df=e^xdx$

$dg=\cos\left(x\right)dx$
$\downarrow$
$g\left(x\right)=\sin\left(x\right)$

$\int_{ }^{ }e^x\sin(x)dx= \\ -e^x\cos(x)+ \overbrace{e^{x}\text{sin}(x)-\int_{ }^{ }e^x\sin(x)dx}$

Solve Elise’s equation for $\int e^x \sin(x)dx$ .

7-202.

Examine the following integrals. Some are easier to evaluate by using the substitution method, while others require integration by parts. Evaluate each integral and briefly describe your method.

$\int 2x^3 \ln(x^4)dx$

$\int x \sin(5x)dx$

$3\int e^{2x}\cos(x)dx$

$\int \sec^3(x) \tan(x)dx$

$\int x^2 \sin(x)dx$

$\int x^3 \sin(x)dx$

7-203.

Sketch the graph of $f(x)=\frac{3x^2+5x-2}{x}$ . Write an equation for its end behavior function. Homework Help ✎

7-204.

$y=f(x)$ satisfies $\frac{dy}{dx}=2x$ and its graph contains the point $(-1,5)$ . Homework Help ✎

Use Euler’s Method to sketch an approximation of $y=f(x)$ over $-1\le x\le1$ . Use $Δx=0.5$ .
Use implicit integration to solve the differential equation.
Compare your answers to parts (a) and (b). Determine the largest difference between the actual $y$ -value and the Euler’s Method $y$ -value for $x=-1$ , $-0.5$ , $0$ , $0.5$ , and $1$ . This largest difference estimates the error of Euler’s Method. If $Δx=0.1$ instead, do you think this error will be larger or smaller?

7-205.

Solve for $x$ if $\frac{dx}{dt}=x+1$ and $x=0$ when $t=2$ . Confirm that your solution is correct by substituting into the differential equation. Homework Help ✎

7-206.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int _ { - 1 } ^ { 3 } \frac { d x } { x }$

$\int \operatorname { tan } ^ { - 1 } ( x ) d x$

$\int \frac { \operatorname { sin } \sqrt { x } d x } { \sqrt { x } }$

$\int _ { \pi / 3 } ^ { \pi / 2 } \operatorname { csc } ^ { 2 } ( x ) d x$

7-207.

Evaluate each of the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow 1 } \frac { \operatorname { ln } ( x ) } { x ^ { 2 } - 1 }$

$\lim\limits _ { x \rightarrow 0 ^ { + } } ( 1 + 5 x ) ^ { 2 / x }$

7-208.

If $(a - 2)x + 2b = 3 + 5x$ is true for all values of $x$ , what are the values of $a$ and $b$ ? Homework Help ✎

7-209.

Multiple Choice: A point is moving along a curve $y = f(x)$ . At the instant when the slope of the curve is $-\frac { 1 } { 3 }$ , the $x$ -coordinate of the moving point is increasing at a rate of $5$ units per second. The rate of change, in units per second, of the $y$ -coordinate of the point is: Homework Help ✎

$-\frac { 5 } { 3 }$

$-\frac { 1 } { 3 }$

$\frac { 1 } { 3 }$

$\frac { 5 } { 3 }$

$\frac { 3 } { 5 }$

Calculus Third Edition