Chapter 5: Q 35. (page 429)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

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

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

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

The formula of integration by parts is $\int u d v = u v - \int v d u$

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

Step 3. Integration by parts.

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

So,

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

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

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