Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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Step 10.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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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 $u_{1}$ and $d u_{1}$ .

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

Step 11

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

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

Step 12

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

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

Step 13

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Split the single integral into multiple integrals.

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

Step 18

Apply the constant rule.

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

Step 19

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

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Step 19.1

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

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Step 19.1.1

Differentiate $2 x$ .

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

Step 19.1.2

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 19.1.3

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 19.1.4

Multiply $2$ by $1$ .

$2$

Step 19.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 20

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

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

Step 21

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

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

Step 22

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

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

Step 23

Simplify.

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

Step 24

Substitute back in for each integration substitution variable.

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Step 24.1

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 24.2

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 25

Simplify.

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Step 25.1

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

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

Step 25.2

Apply the distributive property.

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

Step 25.3

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

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

Step 25.4

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

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Step 25.4.1

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

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

Step 25.4.2

Multiply $2$ by $2$ .

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

Step 25.5

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

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

Step 25.6

Apply the distributive property.

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

Step 25.7

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

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

Step 25.8

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

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Step 25.8.1

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

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

Step 25.8.2

Multiply $2$ by $2$ .

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

Step 26

Reorder terms.

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

Step 27

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

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