Calculus Examples

$\int - 24 x^{5} - 4 x + 15 x^{- 4} d x$

Step 1

Split the single integral into multiple integrals.

$\int - 24 x^{5} d x + \int - 4 x d x + \int 15 x^{- 4} d x$

Step 2

Since $- 24$ is constant with respect to $x$ , move $- 24$ out of the integral.

$- 24 \int x^{5} d x + \int - 4 x d x + \int 15 x^{- 4} d x$

Step 3

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

$- 24 (\frac{1}{6} x^{6} + C) + \int - 4 x d x + \int 15 x^{- 4} d x$

Step 4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 24 (\frac{1}{6} x^{6} + C) - 4 \int x d x + \int 15 x^{- 4} d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 24 (\frac{1}{6} x^{6} + C) - 4 (\frac{1}{2} x^{2} + C) + \int 15 x^{- 4} d x$

Step 6

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$- 24 (\frac{1}{6} x^{6} + C) - 4 (\frac{1}{2} x^{2} + C) + 15 \int x^{- 4} d x$

Step 7

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$- 24 (\frac{1}{6} x^{6} + C) - 4 (\frac{1}{2} x^{2} + C) + 15 (- \frac{1}{3} x^{- 3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$- 24 (\frac{x^{6}}{6} + C) - 4 (\frac{1}{2} x^{2} + C) + 15 (- \frac{1}{3} x^{- 3} + C)$

Step 8.1.2

Combine $\frac{1}{2}$ and $x^{2}$ .

$- 24 (\frac{x^{6}}{6} + C) - 4 (\frac{x^{2}}{2} + C) + 15 (- \frac{1}{3} x^{- 3} + C)$

Step 8.1.3

Combine $x^{- 3}$ and $\frac{1}{3}$ .

$- 24 (\frac{x^{6}}{6} + C) - 4 (\frac{x^{2}}{2} + C) + 15 (- \frac{x^{- 3}}{3} + C)$

Step 8.1.4

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 24 (\frac{x^{6}}{6} + C) - 4 (\frac{x^{2}}{2} + C) + 15 (- \frac{1}{3 x^{3}} + C)$

Step 8.2

Simplify.

$- 4 x^{6} - 2 x^{2} + 15 (- \frac{1}{3 x^{3}}) + C$

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $15$ .

$- 4 x^{6} - 2 x^{2} - 15 \frac{1}{3 x^{3}} + C$

Step 8.3.2

Combine $- 15$ and $\frac{1}{3 x^{3}}$ .

$- 4 x^{6} - 2 x^{2} + \frac{- 15}{3 x^{3}} + C$

Step 8.3.3

Cancel the common factor of $- 15$ and $3$ .

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Step 8.3.3.1

Factor $3$ out of $- 15$ .

$- 4 x^{6} - 2 x^{2} + \frac{3 \cdot - 5}{3 x^{3}} + C$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $3$ out of $3 x^{3}$ .

$- 4 x^{6} - 2 x^{2} + \frac{3 \cdot - 5}{3 (x^{3})} + C$

Step 8.3.3.2.2

Cancel the common factor.

$- 4 x^{6} - 2 x^{2} + \frac{3 \cdot - 5}{3 x^{3}} + C$

Step 8.3.3.2.3

Rewrite the expression.

$- 4 x^{6} - 2 x^{2} + \frac{- 5}{x^{3}} + C$

Step 8.3.4

Move the negative in front of the fraction.

$- 4 x^{6} - 2 x^{2} - \frac{5}{x^{3}} + C$