Calculus Examples

$\int {(x - 5)}^{2} e^{x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = {(x - 5)}^{2}$ and $d v = e^{x}$ .

${(x - 5)}^{2} e^{x} - \int e^{x} (2 x - 10) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 2 x - 10$ and $d v = e^{x}$ .

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - \int e^{x} \cdot 2 d x)$

Step 3

Move $2$ to the left of $e^{x}$ .

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - \int 2 e^{x} d x)$

Step 4

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

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - (2 \int e^{x} d x))$

Step 5

Multiply $2$ by $- 1$ .

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - 2 \int e^{x} d x)$

Step 6

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

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - 2 (e^{x} + C))$

Step 7

Simplify the answer.

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Step 7.1

Rewrite ${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - 2 (e^{x} + C))$ as $(x - 5) (x - 5) e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$ .

$(x - 5) (x - 5) e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$

Step 7.2

Simplify.

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Step 7.2.1

Raise $x - 5$ to the power of $1$ .

${(x - 5)}^{1} (x - 5) e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$

Step 7.2.2

Raise $x - 5$ to the power of $1$ .

${(x - 5)}^{1} {(x - 5)}^{1} e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$

Step 7.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x - 5)}^{1 + 1} e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$

Step 7.2.4

Add $1$ and $1$ .

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x} - 2 e^{x}) + C$

Step 7.3

Simplify.

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Step 7.3.1

Apply the distributive property.

${(x - 5)}^{2} e^{x} - ((2 x - 10) e^{x}) - (- 2 e^{x}) + C$

Step 7.3.2

Multiply $- 2$ by $- 1$ .

${(x - 5)}^{2} e^{x} - (2 x - 10) e^{x} + 2 e^{x} + C$

Step 7.3.3

Reorder factors in $- (2 x - 10) e^{x} + 2 e^{x}$ .

${(x - 5)}^{2} e^{x} - e^{x} (2 x - 10) + 2 e^{x} + C$