Calculus Examples

$8 \cos^{2} (x)$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \cos^{2} (x) d x$

Step 2

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

$8 \int \frac{1 + \cos (2 x)}{2} d x$

Step 3

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

$8 (\frac{1}{2} \int 1 + \cos (2 x) d x)$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{2}$ and $8$ .

$\frac{8}{2} \int 1 + \cos (2 x) d x$

Step 4.2

Cancel the common factor of $8$ and $2$ .

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Step 4.2.1

Factor $2$ out of $8$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.2.2.4

Divide $4$ by $1$ .

$4 \int 1 + \cos (2 x) d x$

Step 5

Split the single integral into multiple integrals.

$4 (\int d x + \int \cos (2 x) d x)$

Step 6

Apply the constant rule.

$4 (x + C + \int \cos (2 x) d x)$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

Rewrite the problem using $u$ and $d u$ .

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

Step 8

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

$4 (x + C + \int \frac{\cos (u)}{2} d u)$

Step 9

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

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

Step 10

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

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

Step 11

Simplify.

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

Step 12

Replace all occurrences of $u$ with $2 x$ .

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Apply the distributive property.

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

Step 13.3

Cancel the common factor of $2$ .

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Step 13.3.1

Factor $2$ out of $4$ .

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

Step 13.3.2

Cancel the common factor.

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

Step 13.3.3

Rewrite the expression.

$4 x + 2 \sin (2 x) + C$