Chapter 5: Q 41. (page 429)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Solve the integration by parts .

We have,

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

and

role="math" localid="1648720139610" $d v = e^{- x} d x v = \int e^{- x} d x v = - \frac{1}{e^{x}}$

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

role="math" localid="1648720506794" $\int \frac{x^{2} + 1}{e^{x}} d x = [(x^{2} + 1) (- \frac{1}{e^{x}}) - \int (- \frac{1}{e^{x}}) d x] = [- \frac{1}{e^{x}} (x^{2} + 1) - \int (- \frac{2 x}{e^{x}}) d x] = [- \frac{1}{e^{x}} (x^{2} + 1) + 2 \int (\frac{x}{e^{x}}) d x]$

Step 3. Integration by parts.

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

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

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

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

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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Solve the integral:∫x2+1exdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

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