Calculus Examples

$\frac{d y}{d x} - 5 x^{2} + 2 = 0$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

Tap for more steps...

Step 1.1.1

Add $5 x^{2}$ to both sides of the equation.

$\frac{d y}{d x} + 2 = 5 x^{2}$

Step 1.1.2

Subtract $2$ from both sides of the equation.

$\frac{d y}{d x} = 5 x^{2} - 2$

Step 1.2

Rewrite the equation.

$d y = (5 x^{2} - 2) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 5 x^{2} - 2 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 5 x^{2} - 2 d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 5 x^{2} d x + \int - 2 d x$

Step 2.3.2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$y + C_{1} = 5 \int x^{2} d x + \int - 2 d x$

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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

Step 2.3.5.3

Reorder terms.

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

Step 2.4

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

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