Calculus Examples

$\frac{d y}{d x} = 8 x^{\frac{5}{7}}$

Step 1

Rewrite the equation.

$d y = 8 x^{\frac{5}{7}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 x^{\frac{5}{7}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 x^{\frac{5}{7}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$y + C_{1} = 8 \int x^{\frac{5}{7}} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{\frac{5}{7}}$ with respect to $x$ is $\frac{7}{12} x^{\frac{12}{7}}$ .

$y + C_{1} = 8 (\frac{7}{12} x^{\frac{12}{7}} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $8 (\frac{7}{12} x^{\frac{12}{7}} + C_{2})$ as $8 (\frac{7}{12}) x^{\frac{12}{7}} + C_{2}$ .

$y + C_{1} = 8 (\frac{7}{12}) x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $8$ and $\frac{7}{12}$ .

$y + C_{1} = \frac{8 \cdot 7}{12} x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2.2

Multiply $8$ by $7$ .

$y + C_{1} = \frac{56}{12} x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2.3

Cancel the common factor of $56$ and $12$ .

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Step 2.3.3.2.3.1

Factor $4$ out of $56$ .

$y + C_{1} = \frac{4 (14)}{12} x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2.3.2

Cancel the common factors.

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Step 2.3.3.2.3.2.1

Factor $4$ out of $12$ .

$y + C_{1} = \frac{4 \cdot 14}{4 \cdot 3} x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2.3.2.2

Cancel the common factor.

$y + C_{1} = \frac{4 \cdot 14}{4 \cdot 3} x^{\frac{12}{7}} + C_{2}$

Step 2.3.3.2.3.2.3

Rewrite the expression.

$y + C_{1} = \frac{14}{3} x^{\frac{12}{7}} + C_{2}$

Step 2.4

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

$y = \frac{14}{3} x^{\frac{12}{7}} + K$