Calculus Examples

$\int {(1 - \cos (x))}^{2} d x$

Step 1

Expand ${(1 - \cos (x))}^{2}$ .

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Step 1.1

Rewrite ${(1 - \cos (x))}^{2}$ as $(1 - \cos (x)) (1 - \cos (x))$ .

$\int (1 - \cos (x)) (1 - \cos (x)) d x$

Step 1.2

Apply the distributive property.

$\int 1 (1 - \cos (x)) - \cos (x) (1 - \cos (x)) d x$

Step 1.3

Apply the distributive property.

$\int 1 \cdot 1 + 1 (- \cos (x)) - \cos (x) (1 - \cos (x)) d x$

Step 1.4

Apply the distributive property.

$\int 1 \cdot 1 + 1 (- \cos (x)) - \cos (x) \cdot 1 - \cos (x) (- \cos (x)) d x$

Step 1.5

Move $\cos (x)$ .

$\int 1 \cdot 1 + 1 \cdot - 1 \cos (x) - 1 \cdot 1 \cos (x) - \cos (x) (- \cos (x)) d x$

Step 1.6

Move $\cos (x)$ .

$\int 1 \cdot 1 + 1 \cdot - 1 \cos (x) - 1 \cdot 1 \cos (x) - 1 \cdot - 1 \cos (x) \cos (x) d x$

Step 1.7

Multiply $1$ by $1$ .

$\int 1 + 1 \cdot - 1 \cos (x) - 1 \cdot 1 \cos (x) - 1 \cdot - 1 \cos (x) \cos (x) d x$

Step 1.8

Multiply $- 1$ by $1$ .

$\int 1 - \cos (x) - 1 \cdot 1 \cos (x) - 1 \cdot - 1 \cos (x) \cos (x) d x$

Step 1.9

Multiply $- 1$ by $1$ .

$\int 1 - \cos (x) - \cos (x) - 1 \cdot - 1 \cos (x) \cos (x) d x$

Step 1.10

Multiply $- 1$ by $- 1$ .

$\int 1 - \cos (x) - \cos (x) + 1 \cos (x) \cos (x) d x$

Step 1.11

Multiply $\cos (x)$ by $1$ .

$\int 1 - \cos (x) - \cos (x) + \cos (x) \cos (x) d x$

Step 1.12

Raise $\cos (x)$ to the power of $1$ .

$\int 1 - \cos (x) - \cos (x) + \cos^{1} (x) \cos (x) d x$

Step 1.13

Raise $\cos (x)$ to the power of $1$ .

$\int 1 - \cos (x) - \cos (x) + \cos^{1} (x) \cos^{1} (x) d x$

Step 1.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 1 - \cos (x) - \cos (x) + {\cos (x)}^{1 + 1} d x$

Step 1.15

Add $1$ and $1$ .

$\int 1 - \cos (x) - \cos (x) + \cos^{2} (x) d x$

Step 1.16

Subtract $\cos (x)$ from $- \cos (x)$ .

$\int 1 - 2 \cos (x) + \cos^{2} (x) d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 5

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

$x + C - 2 (\sin (x) + C) + \int \cos^{2} (x) d x$

Step 6

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

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

Step 7

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

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Apply the constant rule.

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

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $2 x$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify.

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

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

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

Step 16.2

Apply the distributive property.

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

Step 16.3

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

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

Step 16.4

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

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Step 16.4.1

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

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

Step 16.4.2

Multiply $2$ by $2$ .

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

Step 17

Reorder terms.

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