Calculus Examples

\int \cos^{2} (y) d y

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

Step 1

Use the half-angle formula to rewrite

\cos^{2} (y)

$\cos^{2} (y)$ as

\frac{1 + \cos (2 y)}{2}

$\frac{1 + \cos (2 y)}{2}$ .

\int \frac{1 + \cos (2 y)}{2} d y

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

Step 2

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

y

$y$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

\frac{1}{2} \int 1 + \cos (2 y) d y

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

Step 3

Split the single integral into multiple integrals.

\frac{1}{2} (\int d y + \int \cos (2 y) d y)

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

Step 4

Apply the constant rule.

\frac{1}{2} (y + C + \int \cos (2 y) d y)

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

Step 5

Let

u = 2 y

$u = 2 y$ . Then

d u = 2 d y

$d u = 2 d y$ , so

\frac{1}{2} d u = d y

$\frac{1}{2} d u = d y$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 5.1

Let

u = 2 y

$u = 2 y$ . Find

\frac{d u}{d y}

$\frac{d u}{d y}$ .

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Step 5.1.1

Differentiate

2 y

$2 y$ .

\frac{d}{d y} [2 y]

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

Step 5.1.2

Since

2

$2$ is constant with respect to

y

$y$ , the derivative of

2 y

$2 y$ with respect to

y

$y$ is

2 \frac{d}{d y} [y]

$2 \frac{d}{d y} [y]$ .

2 \frac{d}{d y} [y]

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

Step 5.1.3

Differentiate using the Power Rule which states that

\frac{d}{d y} [y^{n}]

$\frac{d}{d y} [y^{n}]$ is

n y^{n - 1}

$n y^{n - 1}$ where

n = 1

$n = 1$ .

2 \cdot 1

$2 \cdot 1$

Step 5.1.4

Multiply

2

$2$ by

1

$1$ .

2

$2$

2

$2$

Step 5.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\frac{1}{2} (y + C + \int \cos (u) \frac{1}{2} d u)

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

\frac{1}{2} (y + C + \int \cos (u) \frac{1}{2} d u)

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

Step 6

Combine

\cos (u)

$\cos (u)$ and

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2} (y + C + \int \frac{\cos (u)}{2} d u)

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

Step 7

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

\frac{1}{2} (y + C + \frac{1}{2} \int \cos (u) d u)

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

Step 8

The integral of

\cos (u)

$\cos (u)$ with respect to

u

$u$ is

\sin (u)

$\sin (u)$ .

\frac{1}{2} (y + C + \frac{1}{2} (\sin (u) + C))

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

Step 9

Simplify.

\frac{1}{2} (y + \frac{1}{2} \sin (u)) + C

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

Step 10

Replace all occurrences of

u

$u$ with

2 y

$2 y$ .

\frac{1}{2} (y + \frac{1}{2} \sin (2 y)) + C

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

Step 11

Simplify.

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Step 11.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

\sin (2 y)

$\sin (2 y)$ .

\frac{1}{2} (y + \frac{\sin (2 y)}{2}) + C

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

Step 11.2

Apply the distributive property.

\frac{1}{2} y + \frac{1}{2} \cdot \frac{\sin (2 y)}{2} + C

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

Step 11.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

y

$y$ .

\frac{y}{2} + \frac{1}{2} \cdot \frac{\sin (2 y)}{2} + C

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

Step 11.4

Multiply

\frac{1}{2} \cdot \frac{\sin (2 y)}{2}

$\frac{1}{2} \cdot \frac{\sin (2 y)}{2}$ .

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Step 11.4.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{\sin (2 y)}{2}

$\frac{\sin (2 y)}{2}$ .

\frac{y}{2} + \frac{\sin (2 y)}{2 \cdot 2} + C

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

Step 11.4.2

Multiply

2

$2$ by

2

$2$ .

\frac{y}{2} + \frac{\sin (2 y)}{4} + C

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

\frac{y}{2} + \frac{\sin (2 y)}{4} + C

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

\frac{y}{2} + \frac{\sin (2 y)}{4} + C

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

Step 12

Reorder terms.

\frac{1}{2} y + \frac{1}{4} \sin (2 y) + C

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