Calculus Examples

$x^{2} d x = 3 y^{2} d y$

Step 1

Rewrite the equation.

$3 y^{2} d y = x^{2} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

Tap for more steps...

Step 2.2.3.1

Rewrite $3 (\frac{1}{3} y^{3} + C_{1})$ as $3 (\frac{1}{3}) y^{3} + C_{1}$ .

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

Step 2.2.3.2

Simplify.

Tap for more steps...

Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.3.2.2.1

Cancel the common factor.

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

Step 2.2.3.2.2.2

Rewrite the expression.

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

Step 2.2.3.2.3

Multiply $y^{3}$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{\frac{1}{3} x^{3} + K}$

Step 3.2

Simplify $\sqrt[3]{\frac{1}{3} x^{3} + K}$ .

Tap for more steps...

Step 3.2.1

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

$y = \sqrt[3]{\frac{x^{3}}{3} + K}$

Step 3.2.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \sqrt[3]{\frac{x^{3}}{3} + K \cdot \frac{3}{3}}$

Step 3.2.3

Simplify terms.

Tap for more steps...

Step 3.2.3.1

Combine $K$ and $\frac{3}{3}$ .

$y = \sqrt[3]{\frac{x^{3}}{3} + \frac{K \cdot 3}{3}}$

Step 3.2.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x^{3} + K \cdot 3}{3}}$

Step 3.2.4

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

$y = \sqrt[3]{\frac{x^{3} + 3 K}{3}}$

Step 3.2.5

Rewrite $\sqrt[3]{\frac{x^{3} + 3 K}{3}}$ as $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{3}}$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{3}}$

Step 3.2.6

Multiply $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$

Step 3.2.7

Combine and simplify the denominator.

Tap for more steps...

Step 3.2.7.1

Multiply $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 3.2.7.2

Raise $\sqrt[3]{3}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 3.2.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 3.2.7.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}}$

Step 3.2.7.5

Rewrite ${\sqrt[3]{3}}^{3}$ as $3$ .

Tap for more steps...

Step 3.2.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3}$ as $3^{\frac{1}{3}}$ .

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

Step 3.2.7.5.2

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

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 3.2.7.5.3

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

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 3.2.7.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.7.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 3.2.7.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{3^{1}}$

Step 3.2.7.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{3}}^{2}}{3}$

Step 3.2.8

Simplify the numerator.

Tap for more steps...

Step 3.2.8.1

Rewrite ${\sqrt[3]{3}}^{2}$ as $\sqrt[3]{3^{2}}$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} \sqrt[3]{3^{2}}}{3}$

Step 3.2.8.2

Raise $3$ to the power of $2$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} \sqrt[3]{9}}{3}$

Step 3.2.9

Simplify with factoring out.

Tap for more steps...

Step 3.2.9.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x^{3} + 3 K) \cdot 9}}{3}$

Step 3.2.9.2

Reorder factors in $\frac{\sqrt[3]{(x^{3} + 3 K) \cdot 9}}{3}$ .

$y = \frac{\sqrt[3]{9 (x^{3} + 3 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{9 (x^{3} + K)}}{3}$