Calculus Examples

$\frac{d y}{d x} = \sin^{2} (x)$

Step 1

Rewrite the equation.

$d y = \sin^{2} (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sin^{2} (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Split the single integral into multiple integrals.

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

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

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

Step 2.3.6

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

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Step 2.3.6.1

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

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Step 2.3.6.1.1

Differentiate $2 x$ .

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

Step 2.3.6.1.2

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 2.3.6.1.3

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 2.3.6.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.6.2

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

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

$y + C_{1} = \frac{1}{2} (x + C_{2} - \frac{1}{2} (\sin (u) + C_{3}))$

Step 2.3.10

Simplify.

$y + C_{1} = \frac{1}{2} (x - \frac{1}{2} \sin (u)) + C_{4}$

Step 2.3.11

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

$y + C_{1} = \frac{1}{2} (x - \frac{1}{2} \sin (2 x)) + C_{4}$

Step 2.3.12

Simplify.

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Step 2.3.12.1

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

$y + C_{1} = \frac{1}{2} (x - \frac{\sin (2 x)}{2}) + C_{4}$

Step 2.3.12.2

Apply the distributive property.

$y + C_{1} = \frac{1}{2} x + \frac{1}{2} (- \frac{\sin (2 x)}{2}) + C_{4}$

Step 2.3.12.3

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

$y + C_{1} = \frac{x}{2} + \frac{1}{2} (- \frac{\sin (2 x)}{2}) + C_{4}$

Step 2.3.12.4

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

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Step 2.3.12.4.1

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

$y + C_{1} = \frac{x}{2} - \frac{\sin (2 x)}{2 \cdot 2} + C_{4}$

Step 2.3.12.4.2

Multiply $2$ by $2$ .

$y + C_{1} = \frac{x}{2} - \frac{\sin (2 x)}{4} + C_{4}$

Step 2.3.13

Reorder terms.

$y + C_{1} = \frac{1}{2} x - \frac{1}{4} \sin (2 x) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = \frac{1}{2} x - \frac{1}{4} \sin (2 x) + K$