Calculus Examples

$\int \cos^{2} (2 - 5 x) d x$

Step 1

Let $u_{1} = 2 - 5 x$ . Then $d u_{1} = - 5 d x$ , so $- \frac{1}{5} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 - 5 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 - 5 x$ .

$\frac{d}{d x} [2 - 5 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $2 - 5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$

Step 1.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 5 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$0 - 5 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 5 \cdot 1$

Step 1.1.3.3

Multiply $- 5$ by $1$ .

$0 - 5$

Step 1.1.4

Subtract $5$ from $0$ .

$- 5$

Step 1.2

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

$\int \cos^{2} (u_{1}) \frac{1}{- 5} d u_{1}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int \cos^{2} (u_{1}) (- \frac{1}{5}) d u_{1}$

Step 2.2

Combine $\cos^{2} (u_{1})$ and $\frac{1}{5}$ .

$\int - \frac{\cos^{2} (u_{1})}{5} d u_{1}$

Step 3

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

$- \int \frac{\cos^{2} (u_{1})}{5} d u_{1}$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u_{1}$ , move $\frac{1}{5}$ out of the integral.

$- (\frac{1}{5} \int \cos^{2} (u_{1}) d u_{1})$

Step 5

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

$- \frac{1}{5} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1}$

Step 6

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

$- \frac{1}{5} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1})$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$- \frac{1}{2 \cdot 5} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 7.2

Multiply $2$ by $5$ .

$- \frac{1}{10} \int 1 + \cos (2 u_{1}) d u_{1}$

Step 8

Split the single integral into multiple integrals.

$- \frac{1}{10} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1})$

Step 9

Apply the constant rule.

$- \frac{1}{10} (u_{1} + C + \int \cos (2 u_{1}) d u_{1})$

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $2 u_{1}$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

$- \frac{1}{10} (u_{1} + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 11

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

$- \frac{1}{10} (u_{1} + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 12

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

$- \frac{1}{10} (u_{1} + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 13

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

$- \frac{1}{10} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 14

Simplify.

$- \frac{1}{10} (u_{1} + \frac{1}{2} \sin (u_{2})) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{1}$ with $2 - 5 x$ .

$- \frac{1}{10} (2 - 5 x + \frac{1}{2} \sin (u_{2})) + C$

Step 15.2

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

$- \frac{1}{10} (2 - 5 x + \frac{1}{2} \sin (2 u_{1})) + C$

Step 15.3

Replace all occurrences of $u_{1}$ with $2 - 5 x$ .

$- \frac{1}{10} (2 - 5 x + \frac{1}{2} \sin (2 (2 - 5 x))) + C$