Calculus Examples

$\frac{d y}{d x} = \frac{2}{3} x \sqrt{1 - 9 y^{2}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $\frac{1}{\sqrt{1 - 9 y^{2}}}$ .

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{1}{\sqrt{1 - 9 y^{2}}} (\frac{2}{3} x \sqrt{1 - 9 y^{2}})$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2}{3} \cdot \frac{1}{\sqrt{1 - 9 y^{2}}} (x \sqrt{1 - 9 y^{2}})$

Step 1.2.2

Combine.

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 \cdot 1}{3 \sqrt{1 - 9 y^{2}}} (x \sqrt{1 - 9 y^{2}})$

Step 1.2.3

Cancel the common factor of $\sqrt{1 - 9 y^{2}}$ .

Tap for more steps...

Step 1.2.3.1

Factor $\sqrt{1 - 9 y^{2}}$ out of $3 \sqrt{1 - 9 y^{2}}$ .

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 \cdot 1}{\sqrt{1 - 9 y^{2}} \cdot 3} (x \sqrt{1 - 9 y^{2}})$

Step 1.2.3.2

Factor $\sqrt{1 - 9 y^{2}}$ out of $x \sqrt{1 - 9 y^{2}}$ .

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 \cdot 1}{\sqrt{1 - 9 y^{2}} \cdot 3} (\sqrt{1 - 9 y^{2}} x)$

Step 1.2.3.3

Cancel the common factor.

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 \cdot 1}{\sqrt{1 - 9 y^{2}} \cdot 3} (\sqrt{1 - 9 y^{2}} x)$

Step 1.2.3.4

Rewrite the expression.

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 \cdot 1}{3} x$

Step 1.2.4

Multiply $2$ by $1$ .

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2}{3} x$

Step 1.2.5

Combine $\frac{2}{3}$ and $x$ .

$\frac{1}{\sqrt{1 - 9 y^{2}}} \frac{d y}{d x} = \frac{2 x}{3}$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{1 - 9 y^{2}}} d y = \frac{2 x}{3} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{1 - 9 y^{2}}} d y = \int \frac{2 x}{3} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Write the expression using exponents.

Tap for more steps...

Step 2.2.1.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{\sqrt{1^{2} - 9 y^{2}}} d y = \int \frac{2 x}{3} d x$

Step 2.2.1.2

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

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

Step 2.2.2

Let $u = 3 y$ . Then $d u = 3 d y$ , so $\frac{1}{3} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.2.1

Let $u = 3 y$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 2.2.2.1.1

Differentiate $3 y$ .

$\frac{d}{d y} [3 y]$

Step 2.2.2.1.2

Since $3$ is constant with respect to $y$ , the derivative of $3 y$ with respect to $y$ is $3 \frac{d}{d y} [y]$ .

$3 \frac{d}{d y} [y]$

Step 2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 2.2.2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{\sqrt{1^{2} - u^{2}}} \cdot \frac{1}{3} d u = \int \frac{2 x}{3} d x$

Step 2.2.3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{\sqrt{1^{2} - u^{2}}} d u = \int \frac{2 x}{3} d x$

Step 2.2.4

The integral of $\frac{1}{\sqrt{1^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (u)$

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

Step 2.2.5

Simplify.

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

Step 2.2.6

Replace all occurrences of $u$ with $3 y$ .

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Multiply $\frac{2}{3}$ by $\frac{1}{2}$ .

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

Step 2.3.3.2.2

Multiply $3$ by $2$ .

$\frac{1}{3} \arcsin (3 y) + C_{1} = \frac{2}{6} x^{2} + C_{2}$

Step 2.3.3.2.3

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 2.3.3.2.3.1

Factor $2$ out of $2$ .

$\frac{1}{3} \arcsin (3 y) + C_{1} = \frac{2 (1)}{6} x^{2} + C_{2}$

Step 2.3.3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.2.3.2.1

Factor $2$ out of $6$ .

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

Step 2.3.3.2.3.2.2

Cancel the common factor.

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

Step 2.3.3.2.3.2.3

Rewrite the expression.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply each term in $\frac{1}{3} \arcsin (3 y) = \frac{1}{3} x^{2} + K$ by $3$ to eliminate the fractions.

Tap for more steps...

Step 3.1.1

Multiply each term in $\frac{1}{3} \arcsin (3 y) = \frac{1}{3} x^{2} + K$ by $3$ .

$\frac{1}{3} \arcsin (3 y) \cdot 3 = \frac{1}{3} x^{2} \cdot 3 + K \cdot 3$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Combine $\frac{1}{3}$ and $\arcsin (3 y)$ .

$\frac{\arcsin (3 y)}{3} \cdot 3 = \frac{1}{3} x^{2} \cdot 3 + K \cdot 3$

Step 3.1.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.2.2.1

Cancel the common factor.

$\frac{\arcsin (3 y)}{3} \cdot 3 = \frac{1}{3} x^{2} \cdot 3 + K \cdot 3$

Step 3.1.2.2.2

Rewrite the expression.

$\arcsin (3 y) = \frac{1}{3} x^{2} \cdot 3 + K \cdot 3$

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Combine $\frac{1}{3}$ and $x^{2}$ .

$\arcsin (3 y) = \frac{x^{2}}{3} \cdot 3 + K \cdot 3$

Step 3.1.3.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.3.1.2.1

Cancel the common factor.

$\arcsin (3 y) = \frac{x^{2}}{3} \cdot 3 + K \cdot 3$

Step 3.1.3.1.2.2

Rewrite the expression.

$\arcsin (3 y) = x^{2} + K \cdot 3$

Step 3.1.3.1.3

Move $3$ to the left of $K$ .

$\arcsin (3 y) = x^{2} + 3 K$

Step 3.2

Take the inverse arcsine of both sides of the equation to extract $y$ from inside the arcsine.

$3 y = \sin (x^{2} + 3 K)$

Step 3.3

Divide each term in $3 y = \sin (x^{2} + 3 K)$ by $3$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $3 y = \sin (x^{2} + 3 K)$ by $3$ .

$\frac{3 y}{3} = \frac{\sin (x^{2} + 3 K)}{3}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\sin (x^{2} + 3 K)}{3}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\sin (x^{2} + 3 K)}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sin (x^{2} + K)}{3}$