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Chapter 5: Q 42. (page 429)

Solve the integral : $鈭� \frac{x}{e^{x^{2} + 1}} d x$

Short Answer

Expert verified

The required answer is $- \frac{2 x}{e^{x}} - \frac{1}{e^{x}} (x^{2} + 1) - 2 e^{- x} + C$ .

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� \frac{x}{e^{x^{2} + 1}} d x$ .

02

Step 2. Solve the integration by parts .

We have,

u = x^{2} + 1 d u = 2 x d x

and

$d v = e^{- x} d x v = 鈭� e^{- x} d x v = - \frac{1}{e^{x}}$

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

$鈭� \frac{x}{e^{x^{2} + 1}} d x = [(x^{2} + 1) (- \frac{1}{e^{x}}) - 鈭� (- \frac{1}{e^{x}}) d x] = (- \frac{1}{e^{x}} (x^{2} + 1) - 鈭� (- \frac{2 x}{e^{x}}) d x) = (- \frac{1}{e^{x}} (x^{2} + 1) + 2 鈭� (\frac{x}{e^{x}}) d x)$

03

Step 3. Integration by parts.

We have,

$u = x d u = d x$

and

$d v = e^{- x} d x v = 鈭� e^{- x} d x v = - \frac{1}{e^{x}}$

So,

$(- \frac{1}{e^{x}} (x^{2} + 1) + 2 (x (- \frac{1}{e^{x}}) - 鈭� (- \frac{1}{e^{x}}) d x)) = (- \frac{1}{e^{x}} (x^{2} + 1) + 2 (- \frac{x}{e^{x}} - 鈭� (- \frac{1}{e^{x}}) d x)) = - \frac{1}{e^{x}} (x^{2} + 1) + 2 鈭� e^{- x} d x = - \frac{2 x}{e^{x}} - \frac{1}{e^{x}} (x^{2} + 1) - 2 e^{- x} + c$

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Most popular questions from this chapter

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

$鈭� \frac{1}{\sqrt{u}} d u$

Consider the integral $鈭� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$ from the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution $u = \sin^{鈭� 1} 鈦� x$ to solve this integral.

(b) Use the trigonometric substitution $x = \sin 鈦� u$ to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

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

$鈭� \sin u d u$

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

Solve given definite integral.

${鈭�}_{鈭� 1}^{1} 鈥� x^{3} \sqrt{9 x^{2} 鈭� 1} d x$

See all solutions

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