Calculus Examples

$(y^{8} - y) \frac{d y}{d x} = x$

Step 1

Rewrite the equation.

$(y^{8} - y) d y = x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{8} - y d y = \int x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y^{8} d y + \int - y d y = \int x d x$

Step 2.2.2

By the Power Rule, the integral of $y^{8}$ with respect to $y$ is $\frac{1}{9} y^{9}$ .

$\frac{1}{9} y^{9} + C_{1} + \int - y d y = \int x d x$

Step 2.2.3

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$\frac{1}{9} y^{9} + C_{1} - \int y d y = \int x d x$

Step 2.2.4

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

$\frac{1}{9} y^{9} + C_{1} - (\frac{1}{2} y^{2} + C_{2}) = \int x d x$

Step 2.2.5

Simplify.

$\frac{1}{9} y^{9} - \frac{1}{2} y^{2} + C_{3} = \int x d x$

Step 2.3

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

$\frac{1}{9} y^{9} - \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + C_{4}$

Step 2.4

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

$\frac{1}{9} y^{9} - \frac{1}{2} y^{2} = \frac{1}{2} x^{2} + K$