Calculus Examples

$\frac{d y}{d x} = 12 \sin^{2} (x) \cos (x)$ , $y (0) = 3$

Step 1

Rewrite the equation.

$d y = 12 \sin^{2} (x) \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 12 \sin^{2} (x) \cos (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 12 \sin^{2} (x) \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$y + C_{1} = 12 \int \sin^{2} (x) \cos (x) d x$

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 $d u$ .

$y + C_{1} = 12 \int u^{2} d u$

Step 2.3.3

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

$y + C_{1} = 12 (\frac{1}{3} u^{3} + C_{2})$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Rewrite $12 (\frac{1}{3} u^{3} + C_{2})$ as $12 (\frac{1}{3}) u^{3} + C_{2}$ .

$y + C_{1} = 12 (\frac{1}{3}) u^{3} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Combine $12$ and $\frac{1}{3}$ .

$y + C_{1} = \frac{12}{3} u^{3} + C_{2}$

Step 2.3.4.2.2

Cancel the common factor of $12$ and $3$ .

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Step 2.3.4.2.2.1

Factor $3$ out of $12$ .

$y + C_{1} = \frac{3 \cdot 4}{3} u^{3} + C_{2}$

Step 2.3.4.2.2.2

Cancel the common factors.

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Step 2.3.4.2.2.2.1

Factor $3$ out of $3$ .

$y + C_{1} = \frac{3 \cdot 4}{3 (1)} u^{3} + C_{2}$

Step 2.3.4.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{3 \cdot 4}{3 \cdot 1} u^{3} + C_{2}$

Step 2.3.4.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{4}{1} u^{3} + C_{2}$

Step 2.3.4.2.2.2.4

Divide $4$ by $1$ .

$y + C_{1} = 4 u^{3} + C_{2}$

Step 2.3.5

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

$y + C_{1} = 4 \sin^{3} (x) + C_{2}$

Step 2.4

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

$y = 4 \sin^{3} (x) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $3$ for $y$ in $y = 4 \sin^{3} (x) + K$ .

$3 = 4 \sin^{3} (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $4 \sin^{3} (0) + K = 3$ .

$4 \sin^{3} (0) + K = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 \sin^{3} (0) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

The exact value of $\sin (0)$ is $0$ .

$4 \cdot 0^{3} + K = 3$

Step 4.2.1.1.2

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 + K = 3$

Step 4.2.1.1.3

Multiply $4$ by $0$ .

$0 + K = 3$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 3$

Step 5

Substitute $3$ for $K$ in $y = 4 \sin^{3} (x) + K$ and simplify.

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Step 5.1

Substitute $3$ for $K$ .

$y = 4 \sin^{3} (x) + 3$