Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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

Step 2.4

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

$\sin (y) = \frac{1}{4} x^{4} - 3 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Simplify $\frac{1}{4} x^{4} - 3 x + K$ .

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Step 3.1.1

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

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

Step 3.1.2

Move $- 3 x$ .

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

Step 3.2

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

$y = \arcsin (\frac{x^{4}}{4} + K - 3 x)$