Calculus Examples

$\frac{d y}{d x} = 5 x^{- 6}$

Step 1

Rewrite the equation.

$d y = 5 x^{- 6} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 5 x^{- 6} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 5 x^{- 6} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$y + C_{1} = 5 \int x^{- 6} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$y + C_{1} = 5 (- \frac{1}{5} x^{- 5} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Simplify.

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Step 2.3.3.1.1

Combine $x^{- 5}$ and $\frac{1}{5}$ .

$y + C_{1} = 5 (- \frac{x^{- 5}}{5} + C_{2})$

Step 2.3.3.1.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y + C_{1} = 5 (- \frac{1}{5 x^{5}} + C_{2})$

Step 2.3.3.2

Simplify.

$y + C_{1} = 5 (- \frac{1}{5 x^{5}}) + C_{2}$

Step 2.3.3.3

Simplify.

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Step 2.3.3.3.1

Multiply $- 1$ by $5$ .

$y + C_{1} = - 5 \frac{1}{5 x^{5}} + C_{2}$

Step 2.3.3.3.2

Combine $- 5$ and $\frac{1}{5 x^{5}}$ .

$y + C_{1} = \frac{- 5}{5 x^{5}} + C_{2}$

Step 2.3.3.3.3

Cancel the common factor of $- 5$ and $5$ .

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Step 2.3.3.3.3.1

Factor $5$ out of $- 5$ .

$y + C_{1} = \frac{5 \cdot - 1}{5 x^{5}} + C_{2}$

Step 2.3.3.3.3.2

Cancel the common factors.

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Step 2.3.3.3.3.2.1

Factor $5$ out of $5 x^{5}$ .

$y + C_{1} = \frac{5 \cdot - 1}{5 (x^{5})} + C_{2}$

Step 2.3.3.3.3.2.2

Cancel the common factor.

$y + C_{1} = \frac{5 \cdot - 1}{5 x^{5}} + C_{2}$

Step 2.3.3.3.3.2.3

Rewrite the expression.

$y + C_{1} = \frac{- 1}{x^{5}} + C_{2}$

Step 2.3.3.3.4

Move the negative in front of the fraction.

$y + C_{1} = - \frac{1}{x^{5}} + C_{2}$

Step 2.4

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

$y = - \frac{1}{x^{5}} + K$