Calculus Examples

$\frac{d y}{d x} = 3 \sin^{2} (x) \cos (x)$

Step 1

Rewrite the equation.

$d y = 3 \sin^{2} (x) \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 3 \sin^{2} (x) \cos (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 3 \sin^{2} (x) \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$y + C_{1} = 3 \int \sin^{2} (x) \cos (x) d x$

Step 2.3.2

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 2.3.2.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 2.3.2.2

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

$y + C_{1} = 3 \int u^{2} d u$

Step 2.3.3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$y + C_{1} = 3 (\frac{1}{3} u^{3} + C_{2})$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Rewrite $3 (\frac{1}{3} u^{3} + C_{2})$ as $3 (\frac{1}{3}) u^{3} + C_{2}$ .

$y + C_{1} = 3 (\frac{1}{3}) u^{3} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Combine $3$ and $\frac{1}{3}$ .

$y + C_{1} = \frac{3}{3} u^{3} + C_{2}$

Step 2.3.4.2.2

Cancel the common factor of $3$ .

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Step 2.3.4.2.2.1

Cancel the common factor.

$y + C_{1} = \frac{3}{3} u^{3} + C_{2}$

Step 2.3.4.2.2.2

Rewrite the expression.

$y + C_{1} = 1 u^{3} + C_{2}$

Step 2.3.4.2.3

Multiply $u^{3}$ by $1$ .

$y + C_{1} = u^{3} + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $\sin (x)$ .

$y + C_{1} = \sin^{3} (x) + C_{2}$

Step 2.4

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

$y = \sin^{3} (x) + K$