Calculus Examples

$\frac{d y}{d x} = x \cos (5 x^{2})$ , $y (0) = 8$

Step 1

Rewrite the equation.

$d y = x \cos (5 x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x \cos (5 x^{2}) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x \cos (5 x^{2}) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 5 x^{2}$ . Then $d u = 10 x d x$ , so $\frac{1}{10} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 5 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $5 x^{2}$ .

$\frac{d}{d x} [5 x^{2}]$

Step 2.3.1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$5 (2 x)$

Step 2.3.1.1.4

Multiply $2$ by $5$ .

$10 x$

Step 2.3.1.2

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

$y + C_{1} = \int \cos (u) \frac{1}{10} d u$

Step 2.3.2

Combine $\cos (u)$ and $\frac{1}{10}$ .

$y + C_{1} = \int \frac{\cos (u)}{10} d u$

Step 2.3.3

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$y + C_{1} = \frac{1}{10} \int \cos (u) d u$

Step 2.3.4

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

$y + C_{1} = \frac{1}{10} (\sin (u) + C_{2})$

Step 2.3.5

Simplify.

$y + C_{1} = \frac{1}{10} \sin (u) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $5 x^{2}$ .

$y + C_{1} = \frac{1}{10} \sin (5 x^{2}) + C_{2}$

Step 2.4

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

$y = \frac{1}{10} \sin (5 x^{2}) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $8$ for $y$ in $y = \frac{1}{10} \sin (5 x^{2}) + K$ .

$8 = \frac{1}{10} \sin (5 \cdot 0^{2}) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{10} \cdot \sin (5 \cdot 0^{2}) + K = 8$ .

$\frac{1}{10} \cdot \sin (5 \cdot 0^{2}) + K = 8$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\frac{1}{10} \cdot \sin (5 \cdot 0^{2}) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

$\frac{1}{10} \cdot \sin (5 \cdot 0) + K = 8$

Step 4.2.1.1.2

Multiply $5$ by $0$ .

$\frac{1}{10} \cdot \sin (0) + K = 8$

Step 4.2.1.1.3

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

$\frac{1}{10} \cdot 0 + K = 8$

Step 4.2.1.1.4

Multiply $\frac{1}{10}$ by $0$ .

$0 + K = 8$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 8$

Step 5

Substitute $8$ for $K$ in $y = \frac{1}{10} \sin (5 x^{2}) + K$ and simplify.

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Step 5.1

Substitute $8$ for $K$ .

$y = \frac{1}{10} \sin (5 x^{2}) + 8$

Step 5.2

Combine $\frac{1}{10}$ and $\sin (5 x^{2})$ .

$y = \frac{\sin (5 x^{2})}{10} + 8$