Chapter 5: Q 33. (page 429)

Solve the integral: $\int x^{2} e^{3 x} d x$ .

Short Answer

Expert verified

The required answer is $\frac{x^{2} e^{3 x}}{3} - \frac{2}{9} x e^{3 x} + \frac{2}{27} e^{3 x} + c$ .

Step by step solution

Step 1. Given information.

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

Step 2. Solve the integration by parts .

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

localid="1648652451162">u=x2du=2xdx

and

$d v = e^{3 x} d x v = \int e^{3 x} d x v = \frac{e^{3 x}}{3}$

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

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

Step 3. Integration by parts.

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

$u = x d u = d x$

and

$d v = e^{3 x} d x v = \int e^{3 x} d x v = \frac{e^{3 x}}{3}$

So,

$\frac{1}{3} x^{2} e^{3 x} - \frac{2}{3} \int x \frac{e^{3 x}}{3} d x = \frac{1}{3} x^{2} e^{3 x} - \frac{2}{3} [\frac{1}{3} x e^{3 x} - \frac{1}{3} \int e^{3 x} d x] = \frac{1}{3} x^{2} e^{3 x} - \frac{2}{9} x e^{3 x} + \frac{2}{27} e^{3 x} + c$

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Solve the integral: $\int x^{2} e^{3 x} d x$ .

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:∫x2e3xdx.

Short Answer

Step by step solution

Step 1. Given information.

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One App. One Place for Learning.

Most popular questions from this chapter

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Solve the integral: $\int x^{2} e^{3 x} d x$ .