Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

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

Step 5

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

$5 x + C + 5 (\frac{1}{2} x^{2} + C) + \int 6 x^{- 3} d x$

Step 6

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

$5 x + C + 5 (\frac{1}{2} x^{2} + C) + 6 \int x^{- 3} d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.2

Simplify.

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $6$ .

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

Step 8.3.2

Combine $- 6$ and $\frac{1}{2 x^{2}}$ .

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

Step 8.3.3

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

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Step 8.3.3.1

Factor $2$ out of $- 6$ .

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

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

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

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

Step 8.3.3.2.2

Cancel the common factor.

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

Step 8.3.3.2.3

Rewrite the expression.

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

Step 8.3.4

Move the negative in front of the fraction.

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

Step 8.4

Reorder terms.

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