Calculus Examples

$\frac{d y}{d x} = 1 - \frac{8}{3} x^{\frac{5}{3}}$

Step 1

Rewrite the equation.

$d y = (1 - \frac{8}{3} x^{\frac{5}{3}}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 1 - \frac{8}{3} x^{\frac{5}{3}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 1 - \frac{8}{3} x^{\frac{5}{3}} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int d x + \int - \frac{8}{3} x^{\frac{5}{3}} d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = x + C_{2} + \int - \frac{8}{3} x^{\frac{5}{3}} d x$

Step 2.3.3

Since $- \frac{8}{3}$ is constant with respect to $x$ , move $- \frac{8}{3}$ out of the integral.

$y + C_{1} = x + C_{2} - \frac{8}{3} \int x^{\frac{5}{3}} d x$

Step 2.3.4

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$y + C_{1} = x + C_{2} - \frac{8}{3} (\frac{3}{8} x^{\frac{8}{3}} + C_{3})$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

Simplify.

$y + C_{1} = x - \frac{8}{3} \cdot \frac{3}{8} x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2

Simplify.

Tap for more steps...

Step 2.3.5.2.1

Multiply $\frac{3}{8}$ by $\frac{8}{3}$ .

$y + C_{1} = x - \frac{3 \cdot 8}{8 \cdot 3} x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2.2

Multiply $3$ by $8$ .

$y + C_{1} = x - \frac{24}{8 \cdot 3} x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2.3

Multiply $8$ by $3$ .

$y + C_{1} = x - \frac{24}{24} x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2.4

Cancel the common factor of $24$ .

Tap for more steps...

Step 2.3.5.2.4.1

Cancel the common factor.

$y + C_{1} = x - \frac{24}{24} x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2.4.2

Rewrite the expression.

$y + C_{1} = x - 1 \cdot 1 x^{\frac{8}{3}} + C_{4}$

Step 2.3.5.2.5

Multiply $- 1$ by $1$ .

$y + C_{1} = x - x^{\frac{8}{3}} + C_{4}$

Step 2.4

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

$y = x - x^{\frac{8}{3}} + K$