Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

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

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

Step 2.3.6.2

Simplify.

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

Step 2.3.6.3

Reorder terms.

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

Step 2.4

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

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