Calculus Examples

$\frac{d Q}{d x} = \frac{8}{5} x^{\frac{3}{5}}$

Step 1

Rewrite the equation.

$d Q = \frac{8}{5} x^{\frac{3}{5}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d Q = \int \frac{8}{5} x^{\frac{3}{5}} d x$

Step 2.2

Apply the constant rule.

$Q + C_{1} = \int \frac{8}{5} x^{\frac{3}{5}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{8}{5}$ is constant with respect to $x$ , move $\frac{8}{5}$ out of the integral.

$Q + C_{1} = \frac{8}{5} \int x^{\frac{3}{5}} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{\frac{3}{5}}$ with respect to $x$ is $\frac{5}{8} x^{\frac{8}{5}}$ .

$Q + C_{1} = \frac{8}{5} (\frac{5}{8} x^{\frac{8}{5}} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{8}{5} (\frac{5}{8} x^{\frac{8}{5}} + C_{2})$ as $\frac{8}{5} \cdot \frac{5}{8} x^{\frac{8}{5}} + C_{2}$ .

$Q + C_{1} = \frac{8}{5} \cdot \frac{5}{8} x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{8}{5}$ by $\frac{5}{8}$ .

$Q + C_{1} = \frac{8 \cdot 5}{5 \cdot 8} x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2.2

Multiply $8$ by $5$ .

$Q + C_{1} = \frac{40}{5 \cdot 8} x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2.3

Multiply $5$ by $8$ .

$Q + C_{1} = \frac{40}{40} x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2.4

Cancel the common factor of $40$ .

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Step 2.3.3.2.4.1

Cancel the common factor.

$Q + C_{1} = \frac{40}{40} x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2.4.2

Rewrite the expression.

$Q + C_{1} = 1 x^{\frac{8}{5}} + C_{2}$

Step 2.3.3.2.5

Multiply $x^{\frac{8}{5}}$ by $1$ .

$Q + C_{1} = x^{\frac{8}{5}} + C_{2}$

Step 2.4

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

$Q = x^{\frac{8}{5}} + K$