Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

Multiply $- 1$ by $- 1$ .

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

Step 9

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

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

Step 10

Multiply by $1$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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