Calculus Examples

$\frac{d y}{d x} = 2 x^{\frac{5}{8}}$

Step 1

Rewrite the equation.

$d y = 2 x^{\frac{5}{8}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 2 x^{\frac{5}{8}} d x$

Step 2.2

Apply the constant rule.

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

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.

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

Step 2.3.2

By the Power Rule, the integral of $x^{\frac{5}{8}}$ with respect to $x$ is $\frac{8}{13} x^{\frac{13}{8}}$ .

$y + C_{1} = 2 (\frac{8}{13} x^{\frac{13}{8}} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{8}{13} x^{\frac{13}{8}} + C_{2})$ as $2 (\frac{8}{13}) x^{\frac{13}{8}} + C_{2}$ .

$y + C_{1} = 2 (\frac{8}{13}) x^{\frac{13}{8}} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $2$ and $\frac{8}{13}$ .

$y + C_{1} = \frac{2 \cdot 8}{13} x^{\frac{13}{8}} + C_{2}$

Step 2.3.3.2.2

Multiply $2$ by $8$ .

$y + C_{1} = \frac{16}{13} x^{\frac{13}{8}} + C_{2}$

Step 2.4

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

$y = \frac{16}{13} x^{\frac{13}{8}} + K$