Calculus Examples

$6 \sqrt{x} - \sqrt[6]{x}$

Step 1

Write $6 \sqrt{x} - \sqrt[6]{x}$ as a function.

$f (x) = 6 \sqrt{x} - \sqrt[6]{x}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 6 \sqrt{x} - \sqrt[6]{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int 6 \sqrt{x} d x + \int - \sqrt[6]{x} d x$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \sqrt{x} d x + \int - \sqrt[6]{x} d x$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$6 \int x^{\frac{1}{2}} d x + \int - \sqrt[6]{x} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$6 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - \sqrt[6]{x} d x$

Step 8

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

$6 (\frac{2}{3} x^{\frac{3}{2}} + C) - \int \sqrt[6]{x} d x$

Step 9

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$6 (\frac{2}{3} x^{\frac{3}{2}} + C) - \int x^{\frac{1}{6}} d x$

Step 10

By the Power Rule, the integral of $x^{\frac{1}{6}}$ with respect to $x$ is $\frac{6}{7} x^{\frac{7}{6}}$ .

$6 (\frac{2}{3} x^{\frac{3}{2}} + C) - (\frac{6}{7} x^{\frac{7}{6}} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$6 (\frac{2}{3}) x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $6$ and $\frac{2}{3}$ .

$\frac{6 \cdot 2}{3} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.2

Multiply $6$ by $2$ .

$\frac{12}{3} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.3

Cancel the common factor of $12$ and $3$ .

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Step 11.2.3.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.3.2

Cancel the common factors.

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Step 11.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 4}{3 (1)} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 \cdot 1} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.3.2.3

Rewrite the expression.

$\frac{4}{1} x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 11.2.3.2.4

Divide $4$ by $1$ .

$4 x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = 6 \sqrt{x} - \sqrt[6]{x}$ .

$F (x) =$ $4 x^{\frac{3}{2}} - \frac{6}{7} x^{\frac{7}{6}} + C$