Calculus Examples

$\int (\frac{x^{6} + 5 x^{9} - 4}{x^{3}}) d x$

Step 1

Remove parentheses.

$\int \frac{x^{6} + 5 x^{9} - 4}{x^{3}} d x$

Step 2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{6} + 5 x^{9} - 4) {(x^{3})}^{- 1} d x$

Step 3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.1

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

$\int (x^{6} + 5 x^{9} - 4) x^{3 \cdot - 1} d x$

Step 3.2

Multiply $3$ by $- 1$ .

$\int (x^{6} + 5 x^{9} - 4) x^{- 3} d x$

Step 4

Expand $(x^{6} + 5 x^{9} - 4) x^{- 3}$ .

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Step 4.1

Apply the distributive property.

$\int (x^{6} + 5 x^{9}) x^{- 3} - 4 x^{- 3} d x$

Step 4.2

Apply the distributive property.

$\int x^{6} x^{- 3} + 5 x^{9} x^{- 3} - 4 x^{- 3} d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{6 - 3} + 5 x^{9} x^{- 3} - 4 x^{- 3} d x$

Step 4.4

Subtract $3$ from $6$ .

$\int x^{3} + 5 x^{9} x^{- 3} - 4 x^{- 3} d x$

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{3} + 5 x^{9 - 3} - 4 x^{- 3} d x$

Step 4.6

Subtract $3$ from $9$ .

$\int x^{3} + 5 x^{6} - 4 x^{- 3} d x$

Step 4.7

Reorder $x^{3}$ and $5 x^{6}$ .

$\int 5 x^{6} + x^{3} - 4 x^{- 3} d x$

Step 5

Split the single integral into multiple integrals.

$\int 5 x^{6} d x + \int x^{3} d x + \int - 4 x^{- 3} d x$

Step 6

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

$5 \int x^{6} d x + \int x^{3} d x + \int - 4 x^{- 3} d x$

Step 7

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

$5 (\frac{1}{7} x^{7} + C) + \int x^{3} d x + \int - 4 x^{- 3} d x$

Step 8

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

$5 (\frac{1}{7} x^{7} + C) + \frac{1}{4} x^{4} + C + \int - 4 x^{- 3} d x$

Step 9

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

$5 (\frac{x^{7}}{7} + C) + \frac{1}{4} x^{4} + C + \int - 4 x^{- 3} d x$

Step 10

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

$5 (\frac{x^{7}}{7} + C) + \frac{1}{4} x^{4} + C - 4 \int x^{- 3} d x$

Step 11

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

$5 (\frac{x^{7}}{7} + C) + \frac{1}{4} x^{4} + C - 4 (- \frac{1}{2} x^{- 2} + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

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

$5 (\frac{x^{7}}{7} + C) + \frac{1}{4} x^{4} + C - 4 (- \frac{x^{- 2}}{2} + C)$

Step 12.1.2

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

$5 (\frac{x^{7}}{7} + C) + \frac{1}{4} x^{4} + C - 4 (- \frac{1}{2 x^{2}} + C)$

Step 12.2

Simplify.

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} - 4 (- \frac{1}{2 x^{2}}) + C$

Step 12.3

Simplify.

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Step 12.3.1

Multiply $- 1$ by $- 4$ .

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + 4 \frac{1}{2 x^{2}} + C$

Step 12.3.2

Combine $4$ and $\frac{1}{2 x^{2}}$ .

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + \frac{4}{2 x^{2}} + C$

Step 12.3.3

Cancel the common factor of $4$ and $2$ .

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Step 12.3.3.1

Factor $2$ out of $4$ .

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + \frac{2 \cdot 2}{2 x^{2}} + C$

Step 12.3.3.2

Cancel the common factors.

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Step 12.3.3.2.1

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

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + \frac{2 \cdot 2}{2 (x^{2})} + C$

Step 12.3.3.2.2

Cancel the common factor.

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + \frac{2 \cdot 2}{2 x^{2}} + C$

Step 12.3.3.2.3

Rewrite the expression.

$\frac{5 x^{7}}{7} + \frac{x^{4}}{4} + \frac{2}{x^{2}} + C$

Step 12.4

Reorder terms.

$\frac{5}{7} x^{7} + \frac{1}{4} x^{4} + \frac{2}{x^{2}} + C$