Calculus Examples

$\cos^{2} (2 x)$

Step 1

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

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Step 1.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

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

$2 \frac{d}{d x} [x]$

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \cos^{2} (u_{1}) \frac{1}{2} d u_{1}$

Step 2

Combine $\cos^{2} (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\cos^{2} (u_{1})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int \cos^{2} (u_{1}) d u_{1}$

Step 4

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$\frac{1}{2} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1}$

Step 5

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

$\frac{1}{2} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1})$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 6.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1})$

Step 8

Apply the constant rule.

$\frac{1}{4} (u_{1} + C + \int \cos (2 u_{1}) d u_{1})$

Step 9

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 9.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 9.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 9.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 9.1.3

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

$2 \cdot 1$

Step 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

$\frac{1}{4} (u_{1} + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 10

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{4} (u_{1} + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 11

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

$\frac{1}{4} (u_{1} + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 12

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{4} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 13

Simplify.

$\frac{1}{4} (u_{1} + \frac{1}{2} \sin (u_{2})) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

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

$\frac{1}{4} (2 x + \frac{1}{2} \sin (u_{2})) + C$

Step 14.2

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

$\frac{1}{4} (2 x + \frac{1}{2} \sin (2 u_{1})) + C$

Step 14.3

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

$\frac{1}{4} (2 x + \frac{1}{2} \sin (2 (2 x))) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $2$ by $2$ .

$\frac{1}{4} (2 x + \frac{1}{2} \sin (4 x)) + C$

Step 15.1.2

Combine $\frac{1}{2}$ and $\sin (4 x)$ .

$\frac{1}{4} (2 x + \frac{\sin (4 x)}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{1}{4} (2 x) + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.3

Cancel the common factor of $2$ .

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Step 15.3.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 x) + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.3.2

Factor $2$ out of $2 x$ .

$\frac{1}{2 (2)} (2 (x)) + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.3.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 x) + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.3.4

Rewrite the expression.

$\frac{1}{2} x + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.4

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

$\frac{x}{2} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 15.5

Multiply $\frac{1}{4} \cdot \frac{\sin (4 x)}{2}$ .

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Step 15.5.1

Multiply $\frac{1}{4}$ by $\frac{\sin (4 x)}{2}$ .

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

Step 15.5.2

Multiply $4$ by $2$ .

$\frac{x}{2} + \frac{\sin (4 x)}{8} + C$

Step 16

Reorder terms.

$\frac{1}{2} x + \frac{1}{8} \sin (4 x) + C$