Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1 + \sin (u_{1})}{\cos (u_{1})}$ by $\frac{1}{2}$ .

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

Step 3.2

Move $2$ to the left of $\cos (u_{1})$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $\sin (u_{1})$ .

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

Step 5.1.2

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

$\cos (u_{1})$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $1$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Multiply $1 + u_{2}$ by $1$ .

$\frac{1}{2} \int 1 + u_{2} d u_{2}$

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

Replace all occurrences of $u_{2}$ with $\sin (u_{1})$ .

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

Step 11.2

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

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

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

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

Step 12.4

Combine.

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

Step 12.5

Simplify each term.

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Step 12.5.1

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

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

Step 12.5.2

Multiply $2$ by $2$ .

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

Step 13

Reorder terms.

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