Calculus Examples

$\int (- 3 \sqrt{x} - 6 \sqrt[4]{x}) d x$

Step 1

Remove parentheses.

$\int - 3 \sqrt{x} - 6 \sqrt[4]{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int - 3 \sqrt{x} d x + \int - 6 \sqrt[4]{x} d x$

Step 3

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

$- 3 \int \sqrt{x} d x + \int - 6 \sqrt[4]{x} d x$

Step 4

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

$- 3 \int x^{\frac{1}{2}} d x + \int - 6 \sqrt[4]{x} d x$

Step 5

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

$- 3 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - 6 \sqrt[4]{x} d x$

Step 6

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

$- 3 (\frac{2}{3} x^{\frac{3}{2}} + C) - 6 \int \sqrt[4]{x} d x$

Step 7

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

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

Step 8

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

$- 3 (\frac{2}{3} x^{\frac{3}{2}} + C) - 6 (\frac{4}{5} x^{\frac{5}{4}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$- 3 (\frac{2}{3}) x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{- 3 \cdot 2}{3} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.2

Multiply $- 3$ by $2$ .

$\frac{- 6}{3} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.3

Cancel the common factor of $- 6$ and $3$ .

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Step 9.2.3.1

Factor $3$ out of $- 6$ .

$\frac{3 \cdot - 2}{3} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 2}{3 (1)} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 2}{3 \cdot 1} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{- 2}{1} x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.3.2.4

Divide $- 2$ by $1$ .

$- 2 x^{\frac{3}{2}} - 6 (\frac{4}{5}) x^{\frac{5}{4}} + C$

Step 9.2.4

Combine $- 6$ and $\frac{4}{5}$ .

$- 2 x^{\frac{3}{2}} + \frac{- 6 \cdot 4}{5} x^{\frac{5}{4}} + C$

Step 9.2.5

Multiply $- 6$ by $4$ .

$- 2 x^{\frac{3}{2}} + \frac{- 24}{5} x^{\frac{5}{4}} + C$

Step 9.2.6

Move the negative in front of the fraction.

$- 2 x^{\frac{3}{2}} - \frac{24}{5} x^{\frac{5}{4}} + C$