Chapter 5: Q 45. (page 429)

Solve the integral: $\int {(x - e^{x})}^{2} d x$

Short Answer

Expert verified

The required answer is $\frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + \frac{1}{2} e^{2 x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is $\int {(x - e^{x})}^{2} d x$ .

Step 2. Solve the integration by parts .

We have,

$\int {(x - e^{x})}^{2} d x = \int (x^{2} - 2 x e^{x} + e^{2 x}) d x = \frac{x^{3}}{3} - (2 \int x e^{x} d x) + \int e^{2 x} d x$

Step 3. Integration by parts.

We have,

$u = x d u = d x$

and

$d v = e^{x} d x v = \int e^{x} d x v = e^{x}$

The formula of integration by parts is $\int u d v = u v - \int v d u$
$\frac{x^{3}}{3} - (2 (x e^{x} - \int e^{x} d x)) + \int e^{2 x} d x = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + \int e^{2 x} d x = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + (\frac{1}{2} \int e^{2 x} d x) = \frac{x^{3}}{3} - 2 x e^{x} + 2 e^{x} + \frac{1}{2} e^{2 x} + c$

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Solve the integral: $\int {(x - e^{x})}^{2} d x$

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

Step 1. Given information.

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One App. One Place for Learning.

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Solve the integral: $\int {(x - e^{x})}^{2} d x$