Calculus Examples

$\sin^{2} (θ)$

Step 1

Write $\sin^{2} (θ)$ as a function.

$f (θ) = \sin^{2} (θ)$

Step 2

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

$F (θ) = \int f (θ) d θ$

Step 3

Set up the integral to solve.

$F (θ) = \int \sin^{2} (θ) d θ$

Step 4

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

$\int \frac{1 - \cos (2 θ)}{2} d θ$

Step 5

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

$\frac{1}{2} \int 1 - \cos (2 θ) d θ$

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $2 θ$ .

$\frac{d}{d θ} [2 θ]$

Step 9.1.2

Since $2$ is constant with respect to $θ$ , the derivative of $2 θ$ with respect to $θ$ is $2 \frac{d}{d θ} [θ]$ .

$2 \frac{d}{d θ} [θ]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d θ} [θ^{n}]$ is $n θ^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

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

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

Step 15.2

Apply the distributive property.

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

Step 15.3

Combine $\frac{1}{2}$ and $θ$ .

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

Step 15.4

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

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Step 15.4.1

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

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

Step 15.4.2

Multiply $2$ by $2$ .

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

Step 16

Reorder terms.

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

Step 17

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

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