Calculus Examples

$f (x) = 5 \sqrt[4]{x} - \frac{3}{x}$

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.

$F (x) = \int 5 \sqrt[4]{x} - \frac{3}{x} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 \sqrt[4]{x} d x + \int - \frac{3}{x} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sqrt[4]{x} d x + \int - \frac{3}{x} d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$5 \int x^{\frac{1}{4}} d x + \int - \frac{3}{x} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) + \int - \frac{3}{x} d x$

Step 7

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

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) - \int \frac{3}{x} d x$

Step 8

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

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) - (3 \int \frac{1}{x} d x)$

Step 9

Simplify.

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Step 9.1

Combine $\frac{4}{5}$ and $x^{\frac{5}{4}}$ .

$5 (\frac{4 x^{\frac{5}{4}}}{5} + C) - (3 \int \frac{1}{x} d x)$

Step 9.2

Multiply $3$ by $- 1$ .

$5 (\frac{4 x^{\frac{5}{4}}}{5} + C) - 3 \int \frac{1}{x} d x$

Step 10

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

$5 (\frac{4 x^{\frac{5}{4}}}{5} + C) - 3 (\ln (| x |) + C)$

Step 11

Simplify.

$4 x^{\frac{5}{4}} - 3 \ln (| x |) + C$

Step 12

The answer is the antiderivative of the function $f (x) = 5 \sqrt[4]{x} - \frac{3}{x}$ .

$F (x) =$ $4 x^{\frac{5}{4}} - 3 \ln (| x |) + C$