Calculus Examples

$5 y^{4} \frac{d y}{d x} = 2 x$

Step 1

Rewrite the equation.

$5 y^{4} d y = 2 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

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

Step 2.2.2

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $5 (\frac{1}{5} y^{5} + C_{1})$ as $5 (\frac{1}{5}) y^{5} + C_{1}$ .

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $5$ and $\frac{1}{5}$ .

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

Step 2.2.3.2.2

Cancel the common factor of $5$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

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

Step 2.2.3.2.2.2

Rewrite the expression.

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

Step 2.2.3.2.3

Multiply $y^{5}$ by $1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

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

$y^{5} + C_{1} = 2 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

$y^{5} + C_{1} = 2 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $2$ and $\frac{1}{2}$ .

$y^{5} + C_{1} = \frac{2}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$y^{5} + C_{1} = \frac{2}{2} x^{2} + C_{2}$

Step 2.3.3.2.2.2

Rewrite the expression.

$y^{5} + C_{1} = 1 x^{2} + C_{2}$

Step 2.3.3.2.3

Multiply $x^{2}$ by $1$ .

$y^{5} + C_{1} = x^{2} + C_{2}$

Step 2.4

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

$y^{5} = x^{2} + K$

Step 3

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

$y = \sqrt[5]{x^{2} + K}$