Algebra Examples

$f (x) = {(\frac{1}{5})}^{x}$

Step 1

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

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

Step 2

The integral of ${(\frac{1}{5})}^{x}$ with respect to $x$ is $\frac{{(\frac{1}{5})}^{x}}{\ln (\frac{1}{5})}$ .

$\frac{{(\frac{1}{5})}^{x}}{\ln (\frac{1}{5})} + C$

Step 3

Simplify the answer.

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Step 3.1

Apply the product rule to $\frac{1}{5}$ .

$\frac{\frac{1^{x}}{5^{x}}}{\ln (\frac{1}{5})} + C$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$\frac{\frac{1}{5^{x}}}{\ln (\frac{1}{5})} + C$

Step 3.2.2

Rewrite $\frac{\frac{1}{5^{x}}}{\ln (\frac{1}{5})}$ as a product.

$\frac{1}{5^{x}} \cdot \frac{1}{\ln (\frac{1}{5})} + C$

Step 3.2.3

Multiply $\frac{1}{5^{x}}$ by $\frac{1}{\ln (\frac{1}{5})}$ .

$\frac{1}{5^{x} \ln (\frac{1}{5})} + C$

Step 4

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = \frac{1}{5^{x} \ln (\frac{1}{5})} + C$