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Chapter 5: Q 60. (page 429)

Solve the integral: $鈭� e^{x} \ln (e^{2 x}) d x$

Short Answer

Expert verified

The required answer is $2 x e^{x} - 2 e^{x} + c$ .

Step by step solution

01

Step 1. Given information.

We have given integral is

$鈭� e^{x} \ln (e^{2 x}) d x = 鈭� 2 x e^{x} d x = 2 鈭� x e^{x} d x$ .

02

Step 2. Solve the integration by parts .

We have,

$u = x d u = d x$

and

$d v = e^{x} d x v = 鈭� e^{x} d x v = e^{x}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$ .

$2 鈭� x e^{x} d x = 2 (x . e^{x} - 鈭� e^{x} d x) = 2 x e^{x} - 2 鈭� e^{x} d x = 2 x e^{x} - 2 e^{x} + c$

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Most popular questions from this chapter

List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?

Find three integrals in Exercises 39鈥�74 that can be solved by using a trigonometric substitution of the form $x = a \tan u$ .

Solve the integral: $鈭� (3 - x) e^{2 x} d x$

Solve the integral

$鈭� \sqrt{3 - x} \sqrt{x - 1} d x$

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� e^{u} d u$

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