Calculus Examples

$\frac{d y}{d x} = \frac{4 x^{3}}{\cos (y)}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\cos (y)$ .

$\cos (y) \frac{d y}{d x} = \cos (y) \frac{4 x^{3}}{\cos (y)}$

Step 1.2

Cancel the common factor of $\cos (y)$ .

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Step 1.2.1

Cancel the common factor.

$\cos (y) \frac{d y}{d x} = \cos (y) \frac{4 x^{3}}{\cos (y)}$

Step 1.2.2

Rewrite the expression.

$\cos (y) \frac{d y}{d x} = 4 x^{3}$

Step 1.3

Rewrite the equation.

$\cos (y) d y = 4 x^{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (y) d y = \int 4 x^{3} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2.3.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $4 (\frac{1}{4} x^{4} + C_{2})$ as $4 (\frac{1}{4}) x^{4} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $4$ and $\frac{1}{4}$ .

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

Step 2.3.3.2.2

Cancel the common factor of $4$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

Multiply $x^{4}$ by $1$ .

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

Step 2.4

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

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

Step 3

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

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