Calculus Examples

$\frac{d y}{d x} = x - x^{2}$

Step 1

Rewrite the equation.

$d y = (x - x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x - x^{2} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.4

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

$y = \frac{1}{2} x^{2} - \frac{1}{3} x^{3} + K$