Chapter 5: Q 42. (page 429)

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

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

Step 1. Given information.

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

Step 2. Solve the integration by parts .

We have,

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

and

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

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

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

Step 3. Integration by parts.

We have,

$u = x d u = d x$

and

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

So,

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

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

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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Short Answer

Step by step solution

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One App. One Place for Learning.

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