Calculus Examples

$\int \frac{1}{x^{6}} - x^{6} - \frac{1}{4} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{1}{x^{6}} d x + \int - x^{6} d x + \int - \frac{1}{4} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x^{6})}^{- 1} d x + \int - x^{6} d x + \int - \frac{1}{4} d x$

Step 2.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{6 \cdot - 1} d x + \int - x^{6} d x + \int - \frac{1}{4} d x$

Step 2.2.2

Multiply $6$ by $- 1$ .

$\int x^{- 6} d x + \int - x^{6} d x + \int - \frac{1}{4} d x$

Step 3

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

$- \frac{1}{5} x^{- 5} + C + \int - x^{6} d x + \int - \frac{1}{4} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{5} x^{- 5} + C - \int x^{6} d x + \int - \frac{1}{4} d x$

Step 5

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$- \frac{1}{5} x^{- 5} + C - (\frac{1}{7} x^{7} + C) + \int - \frac{1}{4} d x$

Step 6

Apply the constant rule.

$- \frac{1}{5} x^{- 5} + C - (\frac{1}{7} x^{7} + C) - \frac{1}{4} x + C$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$- \frac{1}{5} x^{- 5} + C - (\frac{x^{7}}{7} + C) - \frac{1}{4} x + C$

Step 7.2

Simplify.

$- \frac{1}{5 x^{5}} - \frac{x^{7}}{7} - \frac{1}{4} x + C$

Step 8

Reorder terms.

$- \frac{1}{5 x^{5}} - \frac{1}{7} x^{7} - \frac{1}{4} x + C$