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Chapter 5: Q 35. (page 429)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information

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

02

Step 2. Solve the integration by parts .

We have, $u = x, d v = \frac{d x}{e^{x}}$

$u = x d u = d x$

and

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

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

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

03

Step 3. Integration by parts.

We have, $u = - x d u = - d x$

So,

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

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Most popular questions from this chapter

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

Solve the integral: $鈭� \ln \sqrt{x} d x$

Why doesn鈥檛 the definite integral ${鈭�}_{2}^{3} \sqrt{1 - x^{2}} d x$ make sense? (Hint: Think about domains.)

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} 鈭� 4 x 鈭� 8$

Solve given definite integral.

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

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