Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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