Calculus Examples

$\frac{d y}{d t} = 8 x^{- 9}$

Step 1

Rewrite the equation.

$d y = 8 x^{- 9} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 x^{- 9} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 x^{- 9} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$y + C_{1} = 8 x^{- 9} t + C_{2}$

Step 2.3.2

Simplify the answer.

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Step 2.3.2.1

Rewrite $8 x^{- 9} t + C_{2}$ as $8 \frac{1}{x^{9}} t + C_{2}$ .

$y + C_{1} = 8 \frac{1}{x^{9}} t + C_{2}$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

Combine $8$ and $\frac{1}{x^{9}}$ .

$y + C_{1} = \frac{8}{x^{9}} t + C_{2}$

Step 2.3.2.2.2

Combine $\frac{8}{x^{9}}$ and $t$ .

$y + C_{1} = \frac{8 t}{x^{9}} + C_{2}$

Step 2.4

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

$y = \frac{8 t}{x^{9}} + K$