Calculus Examples

$\int_{0}^{\ln (2)} x e^{x} d x$

Step 1

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

$x e^{x}]_{0}^{\ln (2)} - \int_{0}^{\ln (2)} e^{x} d x$

Step 2

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

$x e^{x}]_{0}^{\ln (2)} - (e^{x}]_{0}^{\ln (2)})$

Step 3

Substitute and simplify.

Tap for more steps...

Step 3.1

Evaluate $x e^{x}$ at $\ln (2)$ and at $0$ .

$(\ln (2) e^{\ln (2)}) + 0 e^{0} - (e^{x}]_{0}^{\ln (2)})$

Step 3.2

Evaluate $e^{x}$ at $\ln (2)$ and at $0$ .

$(\ln (2) e^{\ln (2)}) + 0 e^{0} - ((e^{\ln (2)}) - e^{0})$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Exponentiation and log are inverse functions.

$\ln (2) \cdot 2 + 0 e^{0} - ((e^{\ln (2)}) - e^{0})$

Step 3.3.2

Move $2$ to the left of $\ln (2)$ .

$2 \cdot \ln (2) + 0 e^{0} - ((e^{\ln (2)}) - e^{0})$

Step 3.3.3

Anything raised to $0$ is $1$ .

$2 \ln (2) + 0 \cdot 1 - ((e^{\ln (2)}) - e^{0})$

Step 3.3.4

Multiply $0$ by $1$ .

$2 \ln (2) + 0 - ((e^{\ln (2)}) - e^{0})$

Step 3.3.5

Add $2 \ln (2)$ and $0$ .

$2 \ln (2) - ((e^{\ln (2)}) - e^{0})$

Step 3.3.6

Exponentiation and log are inverse functions.

$2 \ln (2) - (2 - e^{0})$

Step 3.3.7

Anything raised to $0$ is $1$ .

$2 \ln (2) - (2 - 1 \cdot 1)$

Step 3.3.8

Multiply $- 1$ by $1$ .

$2 \ln (2) - (2 - 1)$

Step 3.3.9

Subtract $1$ from $2$ .

$2 \ln (2) - 1 \cdot 1$

Step 3.3.10

Multiply $- 1$ by $1$ .

$2 \ln (2) - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 \ln (2) - 1$

Decimal Form:

$0.38629436 \dots$

Step 5