Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 y$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

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

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

Step 12.4

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

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Step 12.4.1

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

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

Step 12.4.2

Multiply $2$ by $2$ .

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

Step 13

Reorder terms.

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