Calculus Examples

$\frac{d y}{d x} = 2 x + 3 x^{2}$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Simplify.

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

Step 2.3.6.2

Simplify.

Tap for more steps...

Step 2.3.6.2.1

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

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

Step 2.3.6.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.6.2.2.1

Cancel the common factor.

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

Step 2.3.6.2.2.2

Rewrite the expression.

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

Step 2.3.6.2.3

Multiply $x^{2}$ by $1$ .

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

Step 2.3.6.2.4

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

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

Step 2.3.6.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.6.2.5.1

Cancel the common factor.

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

Step 2.3.6.2.5.2

Rewrite the expression.

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

Step 2.3.6.2.6

Multiply $x^{3}$ by $1$ .

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

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