7.4.2Can I split an integral into parts?

Integration by Parts

7-187.

With your team, brainstorm a list of the differentiation tools you have developed so far. Then complete parts (a) through (c) below.

Write antiderivative functions for $y = x^2$ and $y = \sin^2(x)\cos(x)$ .
Each antiderivative in part (a) could be found by undoing one of the differentiation tools that you listed. Name the tools that you used.
Try to find the antiderivative of $y = x\ln(x)$ , if possible. What happens?

7-188.

INTEGRATION BY PARTS, The Formula

Evaluating an integral such as $\int x \ln(x)dx$ (whose integrand is a product) requires a method called integration by parts, whose formula can be generated by undoing the Product Rule.

State the Product Rule: If $y = f · g$ , write an equation for $y^\prime$ .
To undo the Product Rule, integrate all terms on both sides of the equation. Use definite integrals with bounds from $a$ to $x$ . The result will have three definite integrals.
Use the Fundamental Theorem of Calculus to simplify $\int _ { a } ^ { x } y ^ { \prime } ( t ) d t$ . Then, in terms of $f$ and $g$ , substitute that result into the equation in part (b).
Notice that there are two integrals in the resulting equation. Solve for one of them to create the integration by parts formula. Compare your results with another team before moving on.

7-189.

INTEGRATION BY PARTS, Evaluating $\int x \ln(x)dx$

Confirm that the equation you wrote in part (c) of problem 7-188 is equivalent to $\int _ { a } ^ { x } f ( t ) d g = f ( x ) g ( x ) - \int _ { a } ^ { x } g ( t ) d f$ .
Notice that $\int _ { a } ^ { x } f ( t ) d g$ , on the left side of the equation given in part (a), represents the integral that you are trying to evaluate. Its factors, $f(t)$ and $dg$ , are related to the terms on the right side of the equation. For example, $g$ and $dg$ are related. What other relationships can you find?
In order to evaluate $\int x \ln(x)dx$ , you need to relate it to $\int _ { a } ^ { x } f ( t ) d g$ . You will need to decide which factor, $x$ or $\ln(x)$ , should be substituted for $f(t)$ and which should be substituted for $dg$ . List four different ways to do this.
For each choice in part (c), determine the corresponding $df$ and $g$ , if possible. Eliminate all choices from the list that are not possible.
You should be down to two choices. Since evaluating $\int x \ln(x)dx$ will require you to evaluate $\int _ { a } ^ { x } f ( t ) d g$ , you want the integral to be easy to antidifferentiate. Decide which of the remaining choices is the best. Then substitute and evaluate.
Differentiate you answer to verify that it is correct.

7-190.

Integrate.

$\int 2 x e ^ { x } d x$

$\int x ^ { 2 } \operatorname { ln } ( x ) d x$

$\int _ { 0 } ^ { 1 } \operatorname { sin } ^ { - 1 } ( x ) d x$

Check your results by differentiating.

7-191.

In your own words, summarize the integration by parts method.

Review and Preview problems below

7-192.

Demonstrate that $\frac { x + 11 } { x ^ { 2 } - 3 x - 4 } = \frac { 3 } { x - 4 } - \frac { 2 } { x + 1 }$ . Then, use this fact to integrate $\int \frac { x + 11 } { x ^ { 2 } - 3 x - 4 } d x$ . Homework Help ✎

7-193.

Compute without a calculator No calculator! Evaluate each definite integral below. If the integral is improper, rewrite it in “proper” limit notation first. One of these integrals will need to be written as the sum of two improper integrals as the first step. Homework Help ✎

$\int _ { 1 } ^ { \infty } \frac { \operatorname { tan } ^ { - 1 } ( x ) d x } { 1 + x ^ { 2 } }$
$\int _ { 0 } ^ { \pi } x \operatorname { cos } ( x ) d x$
$\int _ { 0 } ^ { 4 } \frac { d x } { ( x - 2 ) ^ { 2 / 3 } }$

7-194.

Differentiate each of the following equations. Homework Help ✎

$y =\operatorname { sin } ^ { - 1 } ( \sqrt { 1 - x } )$

$y =\frac { 2 \operatorname { cos } ( x ) } { \sqrt { 1 + x ^ { 2 } } }$

$y = xe^{sec(x)}$

$\ln(y) = x \ln(x)$

7-195.

Scarfin’ Willie, also known affectionately as “Hoover,” always wins the annual hot dog eating contest. But even Willie slows down as he scarfs! In fact, his rate of hot dog consumption is inversely proportional to the number of hot dogs he has eaten.

Ten minutes after the start of the contest, Willie is eating $2$ hot dogs per minute. After $15$ total minutes of eating are allowed, time is called and Willie is once again declared the winner. How many hot dogs has Willie eaten? Homework Help ✎

$\$

7-196.

If $(0, 4)$ is a point on the solution curve, use Euler’s Method to sketch the approximate solution curve for $0 ≤ x ≤ 2$ when $\frac { d y } { d x }= 2(x - y)$ . Use $Δx = 0.5$ . Homework Help ✎

7-197.

No calculator! Evaluate the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ^ { 2 } ) } { x ^ { 2 } }$

$\lim\limits _ { x \rightarrow \infty } \frac { 100 x + 40 x ^ { 2 } - 5 x ^ { 3 } } { 10 x ^ { 3 } + 50 x ^ { 2 } - 100 }$

$\lim\limits _ { x \rightarrow \pi ^ { + } } \operatorname { csc } ( x )$

$\lim\limits _ { h \rightarrow 0 } \frac { e ^ { 2 + h } - e ^ { 2 } } { h }$

$\lim\limits _ { x \rightarrow \infty } \frac { e ^ { x } + x ^ { 3 } } { e ^ { x } + x ^ { 2 } }$

Compute without a calculator

7-198.

If $y = x + \sin(xy)$ , then $\frac { d y } { d x }=$ Homework Help ✎

$1 + \cos(xy)$

$1 + y\cos(xy)$

$\frac { 1 } { 1 - \operatorname { cos } ( x y ) }$

$\frac { 1 } { 1 - x \operatorname { cos } ( x y ) }$

$\frac { 1 + y \operatorname { cos } ( x y ) } { 1 - x \operatorname { cos } ( x y ) }$

Calculus Third Edition