Calculus Examples

$\int (6 \sqrt{x^{3}} + 7 \sqrt[3]{x^{2}}) d x$

Step 1

Remove parentheses.

$\int 6 \sqrt{x^{3}} + 7 \sqrt[3]{x^{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int 6 \sqrt{x^{3}} d x + \int 7 \sqrt[3]{x^{2}} d x$

Step 3

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

$6 \int \sqrt{x^{3}} d x + \int 7 \sqrt[3]{x^{2}} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$6 \int x^{\frac{3}{2}} d x + \int 7 \sqrt[3]{x^{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) + \int 7 \sqrt[3]{x^{2}} d x$

Step 6

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) + 7 \int \sqrt[3]{x^{2}} d x$

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) + 7 \int x^{\frac{2}{3}} d x$

Step 8

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) + 7 (\frac{3}{5} x^{\frac{5}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$6 (\frac{2}{5}) x^{\frac{5}{2}} + 7 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $6$ and $\frac{2}{5}$ .

$\frac{6 \cdot 2}{5} x^{\frac{5}{2}} + 7 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 9.2.2

Multiply $6$ by $2$ .

$\frac{12}{5} x^{\frac{5}{2}} + 7 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 9.2.3

Combine $7$ and $\frac{3}{5}$ .

$\frac{12}{5} x^{\frac{5}{2}} + \frac{7 \cdot 3}{5} x^{\frac{5}{3}} + C$

Step 9.2.4

Multiply $7$ by $3$ .

$\frac{12}{5} x^{\frac{5}{2}} + \frac{21}{5} x^{\frac{5}{3}} + C$