Calculus Examples

$\int \sin^{2} (θ) d θ$

Step 1

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 2

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 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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 6

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 6.1

Let

$u = 2 θ$ . Find

$\frac{d u}{d θ}$ .

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Step 6.1.1

Differentiate

$2 θ$ .

$\frac{d}{d θ} [2 θ]$

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply

$2$ by

$1$ .

$2$

Step 6.2

Rewrite the problem using

$u$ and

$d u$ .

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

Step 7

Combine

$\cos (u)$ and

$\frac{1}{2}$ .

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

Step 8

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 9

The integral of

$\cos (u)$ with respect to

$u$ is

$\sin (u)$ .

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

Step 10

Simplify.

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

Step 11

Replace all occurrences of

$u$ with

$2 θ$ .

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

Step 12

Simplify.

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Step 12.1

Combine

$\sin (2 θ)$ and

$\frac{1}{2}$ .

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Combine

$\frac{1}{2}$ and

$θ$ .

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

Step 12.4

Multiply

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

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Step 12.4.1

Multiply

$\frac{1}{2}$ by

$\frac{\sin (2 θ)}{2}$ .

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

Step 12.4.2

Multiply

$2$ by

$2$ .

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

Step 13

Reorder terms.

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