Calculus Examples

$(y^{3} + \sin (y)) d y = x^{2} d x$

Step 1

Set up an integral on each side.

$\int y^{3} + \sin (y) d y = \int x^{2} d x$

Step 2

Integrate the left side.

Tap for more steps...

Step 2.1

Split the single integral into multiple integrals.

$\int y^{3} d y + \int \sin (y) d y = \int x^{2} d x$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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

Step 3

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

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

Step 4

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

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