Calculus Examples

$\int 2 \sqrt[3]{\sin (2 x)} \cos (2 x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sqrt[3]{\sin (2 x)} \cos (2 x) d x$

Step 2

Let $u_{2} = \sin (2 x)$ . Then $d u_{2} = 2 \cos (2 x) d x$ , so $\frac{1}{2} d u_{2} = \cos (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = \sin (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $\sin (2 x)$ .

$\frac{d}{d x} [\sin (2 x)]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [2 x]$

Step 2.1.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [2 x]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$\cos (2 x) \frac{d}{d x} [2 x]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\cos (2 x) (2 \frac{d}{d x} [x])$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\cos (2 x) (2 \cdot 1)$

Step 2.1.3.3

Simplify the expression.

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Step 2.1.3.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 2.1.3.3.2

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x)$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$2 \int \sqrt[3]{u_{2}} \frac{1}{2} d u_{2}$

Step 3

Combine $\sqrt[3]{u_{2}}$ and $\frac{1}{2}$ .

$2 \int \frac{\sqrt[3]{u_{2}}}{2} d u_{2}$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int \sqrt[3]{u_{2}} d u_{2})$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int \sqrt[3]{u_{2}} d u_{2}$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{2}{2} \int \sqrt[3]{u_{2}} d u_{2}$

Step 5.1.2.2

Rewrite the expression.

$1 \int \sqrt[3]{u_{2}} d u_{2}$

Step 5.1.3

Multiply $\int \sqrt[3]{u_{2}} d u_{2}$ by $1$ .

$\int \sqrt[3]{u_{2}} d u_{2}$

Step 5.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u_{2}}$ as ${u_{2}}^{\frac{1}{3}}$ .

$\int {u_{2}}^{\frac{1}{3}} d u_{2}$

Step 6

By the Power Rule, the integral of ${u_{2}}^{\frac{1}{3}}$ with respect to $u_{2}$ is $\frac{3}{4} {u_{2}}^{\frac{4}{3}}$ .

$\frac{3}{4} {u_{2}}^{\frac{4}{3}} + C$

Step 7

Replace all occurrences of $u_{2}$ with $\sin (2 x)$ .

$\frac{3}{4} {\sin (2 x)}^{\frac{4}{3}} + C$