Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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