Calculus Examples

$\frac{d y}{d x} = 7 x^{\frac{1}{9}}$

Step 1

Rewrite the equation.

$d y = 7 x^{\frac{1}{9}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 7 x^{\frac{1}{9}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 7 x^{\frac{1}{9}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$y + C_{1} = 7 \int x^{\frac{1}{9}} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{\frac{1}{9}}$ with respect to $x$ is $\frac{9}{10} x^{\frac{10}{9}}$ .

$y + C_{1} = 7 (\frac{9}{10} x^{\frac{10}{9}} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $7 (\frac{9}{10} x^{\frac{10}{9}} + C_{2})$ as $7 (\frac{9}{10}) x^{\frac{10}{9}} + C_{2}$ .

$y + C_{1} = 7 (\frac{9}{10}) x^{\frac{10}{9}} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $7$ and $\frac{9}{10}$ .

$y + C_{1} = \frac{7 \cdot 9}{10} x^{\frac{10}{9}} + C_{2}$

Step 2.3.3.2.2

Multiply $7$ by $9$ .

$y + C_{1} = \frac{63}{10} x^{\frac{10}{9}} + C_{2}$

Step 2.4

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

$y = \frac{63}{10} x^{\frac{10}{9}} + K$