Calculus Examples

$\int \sin^{2} (x) d x$

Step 1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\int \sin (x) \sin (x) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (x)$ and $d v = \sin (x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

Multiply $- 1$ by $- 1$ .

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

Step 9

Multiply $\int \cos^{2} (x) d x$ by $1$ .

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

Step 10

Multiply by $1$ .

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

Step 11

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos^{2} (x)$ and $d v = 1$ .

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

Step 12

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

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

Step 13

Multiply $- 2$ by $- 1$ .

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

Step 14

Solving for $\int \sin^{2} (x) d x$ , we find that $\int \sin^{2} (x) d x$ = $\frac{\sin (x) (- \cos (x)) + \cos^{2} (x) x}{- 1}$ .

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

Step 15

Simplify the answer.

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Step 15.1

Simplify.

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Step 15.1.1

Move the negative one from the denominator of $\frac{\sin (x) (- \cos (x)) + \cos^{2} (x) x}{- 1}$ .

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

Step 15.1.2

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

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

Step 15.2

Rewrite $- (\sin (x) (- \cos (x)) + \cos^{2} (x) x) + C$ as $- (- \sin (x) \cos (x) + \cos^{2} (x) x) + C$ .

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