Calculus Examples

$\int 6 x^{5} e^{x} d x$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x^{5} e^{x} d x$

Step 2

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

$6 (x^{5} e^{x} - \int e^{x} (5 x^{4}) d x)$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$6 (x^{5} e^{x} - (5 \int e^{x} (x^{4}) d x))$

Step 4

Multiply $5$ by $- 1$ .

$6 (x^{5} e^{x} - 5 \int e^{x} (x^{4}) d x)$

Step 5

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - \int e^{x} (4 x^{3}) d x))$

Step 6

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - (4 \int e^{x} (x^{3}) d x)))$

Step 7

Multiply $4$ by $- 1$ .

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 \int e^{x} (x^{3}) d x))$

Step 8

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - \int e^{x} (3 x^{2}) d x)))$

Step 9

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - (3 \int e^{x} (x^{2}) d x))))$

Step 10

Multiply $3$ by $- 1$ .

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 \int e^{x} (x^{2}) d x)))$

Step 11

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - \int e^{x} (2 x) d x))))$

Step 12

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - (2 \int e^{x} (x) d x)))))$

Step 13

Multiply $2$ by $- 1$ .

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 \int e^{x} (x) d x))))$

Step 14

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - \int e^{x} d x)))))$

Step 15

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

$6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C))))))$

Step 16

Rewrite $6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C))))))$ as $6 (x^{5} e^{x} - 5 x^{4} e^{x} + 20 x^{3} e^{x} - 60 x^{2} e^{x} + 120 x e^{x} - 120 e^{x}) + C$ .

$6 (x^{5} e^{x} - 5 x^{4} e^{x} + 20 x^{3} e^{x} - 60 x^{2} e^{x} + 120 x e^{x} - 120 e^{x}) + C$