Calculus Examples

$\int (5 x^{- 3} + 6 x) d x$

Step 1

Remove parentheses.

$\int 5 x^{- 3} + 6 x d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 x^{- 3} d x + \int 6 x d x$

Step 3

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

$5 \int x^{- 3} d x + \int 6 x d x$

Step 4

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$5 (- \frac{1}{2} x^{- 2} + C) + \int 6 x d x$

Step 5

Simplify.

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Step 5.1

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

$5 (- \frac{x^{- 2}}{2} + C) + \int 6 x d x$

Step 5.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (- \frac{1}{2 x^{2}} + C) + \int 6 x d x$

Step 6

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$5 (- \frac{1}{2 x^{2}} + C) + 6 \int x d x$

Step 7

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

$5 (- \frac{1}{2 x^{2}} + C) + 6 (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{- 5}{2 x^{2}} + 6 (\frac{1}{2} x^{2}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Move the negative in front of the fraction.

$- \frac{5}{2 x^{2}} + 6 (\frac{1}{2} x^{2}) + C$

Step 8.2.2

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

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

Step 8.2.3

Combine $6$ and $\frac{x^{2}}{2}$ .

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

Step 8.2.4

Cancel the common factor of $6$ and $2$ .

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Step 8.2.4.1

Factor $2$ out of $6 x^{2}$ .

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

Step 8.2.4.2

Cancel the common factors.

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Step 8.2.4.2.1

Factor $2$ out of $2$ .

$- \frac{5}{2 x^{2}} + \frac{2 (3 x^{2})}{2 (1)} + C$

Step 8.2.4.2.2

Cancel the common factor.

$- \frac{5}{2 x^{2}} + \frac{2 (3 x^{2})}{2 \cdot 1} + C$

Step 8.2.4.2.3

Rewrite the expression.

$- \frac{5}{2 x^{2}} + \frac{3 x^{2}}{1} + C$

Step 8.2.4.2.4

Divide $3 x^{2}$ by $1$ .

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