Calculus Examples

$\int x^{4} e^{- x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{4}$ and $d v = e^{- x}$ .

$x^{4} (- e^{- x}) - \int - e^{- x} (4 x^{3}) d x$

Step 2

Multiply $4$ by $- 1$ .

$x^{4} (- e^{- x}) - \int - 4 e^{- x} x^{3} d x$

Step 3

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$x^{4} (- e^{- x}) - (- 4 \int e^{- x} x^{3} d x)$

Step 4

Multiply $- 4$ by $- 1$ .

$x^{4} (- e^{- x}) + 4 \int e^{- x} x^{3} d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = e^{- x}$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) - \int - e^{- x} (3 x^{2}) d x)$

Step 6

Multiply $3$ by $- 1$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) - \int - 3 e^{- x} x^{2} d x)$

Step 7

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) - (- 3 \int e^{- x} x^{2} d x))$

Step 8

Multiply $- 3$ by $- 1$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 \int e^{- x} x^{2} d x)$

Step 9

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{- x}$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) - \int - e^{- x} (2 x) d x))$

Step 10

Multiply $2$ by $- 1$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) - \int - 2 e^{- x} x d x))$

Step 11

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) - (- 2 \int e^{- x} x d x)))$

Step 12

Multiply $- 2$ by $- 1$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 \int e^{- x} x d x))$

Step 13

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- x}$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - \int - e^{- x} d x)))$

Step 14

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - - \int e^{- x} d x)))$

Step 15

Simplify.

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Step 15.1

Multiply $- 1$ by $- 1$ .

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

Step 15.2

Multiply $\int e^{- x} d x$ by $1$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) + \int e^{- x} d x)))$

Step 16

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 16.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 16.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 16.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 16.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 16.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 16.2

Rewrite the problem using $u$ and $d u$ .

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

Step 17

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 18

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - (e^{u} + C))))$

Step 19

Rewrite $x^{4} (- e^{- x}) + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - (e^{u} + C))))$ as $- x^{4} e^{- x} + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - e^{u}))) + C$ .

$- x^{4} e^{- x} + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - e^{u}))) + C$

Step 20

Replace all occurrences of $u$ with $- x$ .

$- x^{4} e^{- x} + 4 (x^{3} (- e^{- x}) + 3 (x^{2} (- e^{- x}) + 2 (x (- e^{- x}) - e^{- x}))) + C$