Chapter 5: Q 34. (page 429)

Solve the integral: $\int (3 - x) e^{2 x} d x$

Short Answer

Expert verified

The required answer is $\frac{e^{2 x}}{2} (- x + 3) + \frac{1}{4} e^{2 x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is $\int (3 - x) e^{2 x} d x$ .

Step 2. Solve the integration by parts .

We have, $u = - x + 3, d v = e^{2 x} d x$

$u = - x + 3 d u = - d x$

and

$d v = e^{2 x} d x v = \int e^{2 x} d x v = \frac{e^{2 x}}{2}$

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

$\int (3 - x) e^{2 x} d x = [(- x + 3) (\frac{e^{2 x}}{2}) - \int - (\frac{e^{2 x}}{2}) d x] = (- x + 3) (\frac{e^{2 x}}{2}) + \frac{1}{2} \int e^{2 x} d x$

Step 3. Integration by parts.

We have,

u = 2 x d u = 2 d x \frac{d u}{2} = d x

So,

$\frac{e^{2 x}}{2} (- x + 3) + \frac{1}{2} \int \frac{e^{2 x}}{2} d u = \frac{e^{2 x}}{2} (- x + 3) + \frac{1}{4} \int e^{2 x} d u = \frac{e^{2 x}}{2} (- x + 3) + \frac{1}{4} e^{2 x} + c$

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Solve the integral: $\int (3 - x) e^{2 x} d x$

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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Short Answer

Step by step solution

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One App. One Place for Learning.

Most popular questions from this chapter

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Solve the integral: $\int (3 - x) e^{2 x} d x$