Chapter 5: Q 60. (page 429)

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

Short Answer

Expert verified

The required answer is $2 x e^{x} - 2 e^{x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is

$\int e^{x} \ln (e^{2 x}) d x = \int 2 x e^{x} d x = 2 \int x e^{x} d x$ .

Step 2. Solve the 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 $\int u d v = u v - \int v d u$ .

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

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Step by step solution

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