Calculus Examples

$\frac{d y}{d x} = - 10 x + 9 x^{2}$

Step 1

Rewrite the equation.

$d y = (- 10 x + 9 x^{2}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int - 10 x + 9 x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 10 x + 9 x^{2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int - 10 x d x + \int 9 x^{2} d x$

Step 2.3.2

Since $- 10$ is constant with respect to $x$ , move $- 10$ out of the integral.

$y + C_{1} = - 10 \int x d x + \int 9 x^{2} d x$

Step 2.3.3

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

$y + C_{1} = - 10 (\frac{1}{2} x^{2} + C_{2}) + \int 9 x^{2} d x$

Step 2.3.4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$y + C_{1} = - 10 (\frac{1}{2} x^{2} + C_{2}) + 9 \int x^{2} d x$

Step 2.3.5

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

$y + C_{1} = - 10 (\frac{1}{2} x^{2} + C_{2}) + 9 (\frac{1}{3} x^{3} + C_{3})$

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Simplify.

$y + C_{1} = - 10 (\frac{1}{2}) x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2

Simplify.

Tap for more steps...

Step 2.3.6.2.1

Combine $- 10$ and $\frac{1}{2}$ .

$y + C_{1} = \frac{- 10}{2} x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $- 10$ and $2$ .

Tap for more steps...

Step 2.3.6.2.2.1

Factor $2$ out of $- 10$ .

$y + C_{1} = \frac{2 \cdot - 5}{2} x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.6.2.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot - 5}{2 (1)} x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 5}{2 \cdot 1} x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{- 5}{1} x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2.2.4

Divide $- 5$ by $1$ .

$y + C_{1} = - 5 x^{2} + 9 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.3

Combine $9$ and $\frac{1}{3}$ .

$y + C_{1} = - 5 x^{2} + \frac{9}{3} x^{3} + C_{4}$

Step 2.3.6.2.4

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 2.3.6.2.4.1

Factor $3$ out of $9$ .

$y + C_{1} = - 5 x^{2} + \frac{3 \cdot 3}{3} x^{3} + C_{4}$

Step 2.3.6.2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.3.6.2.4.2.1

Factor $3$ out of $3$ .

$y + C_{1} = - 5 x^{2} + \frac{3 \cdot 3}{3 (1)} x^{3} + C_{4}$

Step 2.3.6.2.4.2.2

Cancel the common factor.

$y + C_{1} = - 5 x^{2} + \frac{3 \cdot 3}{3 \cdot 1} x^{3} + C_{4}$

Step 2.3.6.2.4.2.3

Rewrite the expression.

$y + C_{1} = - 5 x^{2} + \frac{3}{1} x^{3} + C_{4}$

Step 2.3.6.2.4.2.4

Divide $3$ by $1$ .

$y + C_{1} = - 5 x^{2} + 3 x^{3} + C_{4}$

Step 2.3.7

Reorder terms.

$y + C_{1} = 3 x^{3} - 5 x^{2} + C_{4}$

Step 2.4

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

$y = 3 x^{3} - 5 x^{2} + K$