Calculus Examples

$\int_{0}^{\ln (3)} e^{x + \ln (2)} - e^{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{\ln (3)} e^{x + \ln (2)} d x + \int_{0}^{\ln (3)} - e^{x} d x$

Step 2

Rewrite $\int_{0}^{\ln (3)} e^{x + \ln (2)} d x$ as $\int e^{x} e^{\ln (2)} d x$ .

$\int e^{x} e^{\ln (2)} d x + \int_{0}^{\ln (3)} - e^{x} d x$

Step 3

Rewrite $e^{\ln (2)}$ as $2$ .

$\int e^{x} \cdot 2 d x + \int_{0}^{\ln (3)} - e^{x} d x$

Step 4

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

$\int 2 e^{x} d x + \int_{0}^{\ln (3)} - e^{x} d x$

Step 5

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

$2 \int e^{x} d x + \int_{0}^{\ln (3)} - e^{x} d x$

Step 6

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

$2 (e^{x} + C) + \int_{0}^{\ln (3)} - e^{x} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 (e^{x} + C) - \int_{0}^{\ln (3)} e^{x} d x$

Step 8

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

$2 (e^{x} + C) - (e^{x}]_{0}^{\ln (3)})$

Step 9

Substitute and simplify.

Tap for more steps...

Step 9.1

Evaluate $e^{x}$ at $\ln (3)$ and at $0$ .

$2 (e^{x} + C) - ((e^{\ln (3)}) - e^{0})$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Exponentiation and log are inverse functions.

$2 (e^{x} + C) - (3 - e^{0})$

Step 9.2.2

Anything raised to $0$ is $1$ .

$2 (e^{x} + C) - (3 - 1 \cdot 1)$

Step 9.2.3

Multiply $- 1$ by $1$ .

$2 (e^{x} + C) - (3 - 1)$

Step 9.2.4

Subtract $1$ from $3$ .

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

Step 9.2.5

Multiply $- 1$ by $2$ .

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

Step 10

Simplify.

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