Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Flip sides to get $\frac{d y}{d x}$ on the left side.

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

Step 1.2

Rewrite the equation.

$(1 - y^{5}) d y = (4 + x^{2}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int 1 - y^{5} d y = \int 4 + x^{2} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the single integral into multiple integrals.

$\int d y + \int - y^{5} d y = \int 4 + x^{2} d x$

Step 2.2.2

Apply the constant rule.

$y + C_{1} + \int - y^{5} d y = \int 4 + x^{2} d x$

Step 2.2.3

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

$y + C_{1} - \int y^{5} d y = \int 4 + x^{2} d x$

Step 2.2.4

By the Power Rule, the integral of $y^{5}$ with respect to $y$ is $\frac{1}{6} y^{6}$ .

$y + C_{1} - (\frac{1}{6} y^{6} + C_{2}) = \int 4 + x^{2} d x$

Step 2.2.5

Simplify.

$y - \frac{1}{6} y^{6} + C_{3} = \int 4 + x^{2} d x$

Step 2.2.6

Reorder terms.

$- \frac{1}{6} y^{6} + y + C_{3} = \int 4 + x^{2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$- \frac{1}{6} y^{6} + y + C_{3} = \int 4 d x + \int x^{2} d x$

Step 2.3.2

Apply the constant rule.

$- \frac{1}{6} y^{6} + y + C_{3} = 4 x + C_{4} + \int x^{2} d x$

Step 2.3.3

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

$- \frac{1}{6} y^{6} + y + C_{3} = 4 x + C_{4} + \frac{1}{3} x^{3} + C_{5}$

Step 2.3.4

Simplify.

$- \frac{1}{6} y^{6} + y + C_{3} = 4 x + \frac{1}{3} x^{3} + C_{6}$

Step 2.3.5

Reorder terms.

$- \frac{1}{6} y^{6} + y + C_{3} = \frac{1}{3} x^{3} + 4 x + C_{6}$

Step 2.4

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

$- \frac{1}{6} y^{6} + y = \frac{1}{3} x^{3} + 4 x + K$