Calculus Examples

$\int [5 x + \frac{2}{3 x^{5}}] d x$

Step 1

Remove parentheses.

$\int 5 x + \frac{2}{3 x^{5}} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 x d x + \int \frac{2}{3 x^{5}} d x$

Step 3

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

$5 \int x d x + \int \frac{2}{3 x^{5}} d x$

Step 4

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

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

Step 5

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

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

Step 6

Simplify the expression.

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Step 6.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

Multiply $5$ by $- 1$ .

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

Step 7

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$5 (\frac{x^{2}}{2} + C) + \frac{2}{3} (- \frac{1}{4} x^{- 4} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{5 x^{2}}{2} + \frac{2}{3} (- \frac{1}{4} x^{- 4}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

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

Step 8.2.2

Multiply $\frac{2}{3}$ by $\frac{x^{- 4}}{4}$ .

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

Step 8.2.3

Multiply $3$ by $4$ .

$\frac{5 x^{2}}{2} - \frac{2 x^{- 4}}{12} + C$

Step 8.2.4

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5 x^{2}}{2} - \frac{2}{12 x^{4}} + C$

Step 8.2.5

Cancel the common factor of $2$ and $12$ .

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Step 8.2.5.1

Factor $2$ out of $2$ .

$\frac{5 x^{2}}{2} - \frac{2 (1)}{12 x^{4}} + C$

Step 8.2.5.2

Cancel the common factors.

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Step 8.2.5.2.1

Factor $2$ out of $12 x^{4}$ .

$\frac{5 x^{2}}{2} - \frac{2 (1)}{2 (6 x^{4})} + C$

Step 8.2.5.2.2

Cancel the common factor.

$\frac{5 x^{2}}{2} - \frac{2 \cdot 1}{2 (6 x^{4})} + C$

Step 8.2.5.2.3

Rewrite the expression.

$\frac{5 x^{2}}{2} - \frac{1}{6 x^{4}} + C$

Step 8.3

Reorder terms.

$\frac{5}{2} x^{2} - \frac{1}{6 x^{4}} + C$