Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

Rewrite $3 (3 x^{\frac{1}{3}} + C_{2})$ as $3 \cdot 3 x^{\frac{1}{3}} + C_{2}$ .

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

Step 2.3.3.2

Multiply $3$ by $3$ .

$y + C_{1} = 9 x^{\frac{1}{3}} + C_{2}$

Step 2.4

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

$y = 9 x^{\frac{1}{3}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $- 1$ for $x$ and $- 5$ for $y$ in $y = 9 x^{\frac{1}{3}} + K$ .

$- 5 = 9 {(- 1)}^{\frac{1}{3}} + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $9 {(- 1)}^{\frac{1}{3}} + K = - 5$ .

$9 {(- 1)}^{\frac{1}{3}} + K = - 5$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$9 {({(- 1)}^{3})}^{\frac{1}{3}} + K = - 5$

Step 4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 {(- 1)}^{3 (\frac{1}{3})} + K = - 5$

Step 4.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.3.1

Cancel the common factor.

$9 {(- 1)}^{3 (\frac{1}{3})} + K = - 5$

Step 4.2.3.2

Rewrite the expression.

$9 {(- 1)}^{1} + K = - 5$

Step 4.2.4

Evaluate the exponent.

$9 \cdot - 1 + K = - 5$

Step 4.2.5

Multiply $9$ by $- 1$ .

$- 9 + K = - 5$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

Tap for more steps...

Step 4.3.1

Add $9$ to both sides of the equation.

$K = - 5 + 9$

Step 4.3.2

Add $- 5$ and $9$ .

$K = 4$

Step 5

Substitute $4$ for $K$ in $y = 9 x^{\frac{1}{3}} + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $4$ for $K$ .

$y = 9 x^{\frac{1}{3}} + 4$