Calculus Examples

$f (x) = 3^{x} \ln (3)$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 3^{x} \ln (3) d x$

Step 3

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

$\ln (3) \int 3^{x} d x$

Step 4

The integral of $3^{x}$ with respect to $x$ is $\frac{3^{x}}{\ln (3)}$ .

$\ln (3) (\frac{3^{x}}{\ln (3)} + C)$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Rewrite $\ln (3) (\frac{3^{x}}{\ln (3)} + C)$ as $\ln (3) \frac{1}{\ln (3)} \cdot 3^{x} + C$ .

$\ln (3) \frac{1}{\ln (3)} \cdot 3^{x} + C$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Combine $\ln (3)$ and $\frac{1}{\ln (3)}$ .

$\frac{\ln (3)}{\ln (3)} \cdot 3^{x} + C$

Step 5.2.2

Cancel the common factor of $\ln (3)$ .

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$\frac{\ln (3)}{\ln (3)} \cdot 3^{x} + C$

Step 5.2.2.2

Rewrite the expression.

$1 \cdot 3^{x} + C$

Step 5.2.3

Multiply $3^{x}$ by $1$ .

$3^{x} + C$

Step 6

The answer is the antiderivative of the function $f (x) = 3^{x} \ln (3)$ .

$F (x) =$ $3^{x} + C$