Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2} - 1$ and $d v = e^{x}$ .

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

Step 2

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

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

Step 3

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Rewrite $(x^{2} - 1) e^{x} - 2 (x e^{x} - (e^{x} + C))$ as $x^{2} e^{x} - e^{x} - 2 (x e^{x} - e^{x}) + C$ .

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