Chapter 5: Q 50. (page 429)

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

Short Answer

Expert verified

The required answer is $- \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} - \frac{6 x}{e^{x}} - 6 e^{- x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is

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

Step 2. Solve the integration by parts .

We have,

$u = x^{3} d u = 3 x^{2} d x$

and

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

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

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

Step 3. Integration by parts.

We have,

$u = x^{2} d u = 2 x d x$

and

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

So,

$- \frac{x^{3}}{e^{x}} + 3 (x^{2} (- \frac{1}{e^{x}}) - \int 2 x (- \frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} + 3 (- \frac{x^{2}}{e^{x}} + 2 \int (\frac{x}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 \int \frac{x}{e^{x}} d x$

Step 4. Integration by parts.

We have,

$u = x d u = d x$

and

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

So,

role="math" localid="1648804483219" $- \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 \int \frac{x}{e^{x}} d x = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} - \int (- \frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} + \int (\frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} + \int e^{- x} d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} - \frac{6 x}{e^{x}} - 6 e^{- x} + c$

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

Step 4. Integration by parts.

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Solve the integral:∫x3e-xdx.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

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Step 4. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

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