Calculus Examples

$1 - \sin^{2} (x)$

Step 1

Write $1 - \sin^{2} (x)$ as a function.

$f (x) = 1 - \sin^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 1 - \sin^{2} (x) d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

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

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

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

Differentiate $2 x$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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

Step 17

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

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

Step 18

Simplify.

Tap for more steps...

Step 18.1

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

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

Step 18.2

Apply the distributive property.

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

Step 18.3

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

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

Step 18.4

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

Tap for more steps...

Step 18.4.1

Multiply $- 1$ by $- 1$ .

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

Step 18.4.2

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

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

Step 18.4.3

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

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

Step 18.4.4

Multiply $2$ by $2$ .

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

Step 19

Reorder terms.

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

Step 20

The answer is the antiderivative of the function $f (x) = 1 - \sin^{2} (x)$ .

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