Calculus Examples

$\int \sin^{2} (3 x) \cos (3 x) d x$

Step 1

Let $u_{2} = \sin (3 x)$ . Then $d u_{2} = 3 \cos (3 x) d x$ , so $\frac{1}{3} d u_{2} = \cos (3 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sin (3 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\sin (3 x)$ .

$\frac{d}{d x} [\sin (3 x)]$

Step 1.1.2

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 $g (x) = 3 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [3 x]$

Step 1.1.2.2

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

$\cos (u_{1}) \frac{d}{d x} [3 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$\cos (3 x) \frac{d}{d x} [3 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\cos (3 x) (3 \frac{d}{d x} [x])$

Step 1.1.3.2

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

$\cos (3 x) (3 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $3$ by $1$ .

$\cos (3 x) \cdot 3$

Step 1.1.3.3.2

Move $3$ to the left of $\cos (3 x)$ .

$3 \cos (3 x)$

Step 1.2

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

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

Step 2

Combine ${u_{2}}^{2}$ and $\frac{1}{3}$ .

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

$\frac{1}{3} (\frac{1}{3} {u_{2}}^{3} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{3} (\frac{1}{3} {u_{2}}^{3} + C)$ as $\frac{1}{3} \cdot \frac{1}{3} {u_{2}}^{3} + C$ .

$\frac{1}{3} \cdot \frac{1}{3} {u_{2}}^{3} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{1}{3 \cdot 3} {u_{2}}^{3} + C$

Step 5.2.2

Multiply $3$ by $3$ .

$\frac{1}{9} {u_{2}}^{3} + C$

Step 6

Replace all occurrences of $u_{2}$ with $\sin (3 x)$ .

$\frac{1}{9} \sin^{3} (3 x) + C$