Calculus Examples

$(- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) d x$

Step 1

Write $(- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) d x$ as a function.

$f (x) = (- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) d x$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int (- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) d x d x$

Step 4

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

$d \int (- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) x d x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$d \int (- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3}) x - 15 x d x$

Step 5.2

Apply the distributive property.

$d \int (- 9 x^{\frac{4}{5}} + 3 x^{- 6}) x + 9 x^{3} x - 15 x d x$

Step 5.3

Apply the distributive property.

$d \int - 9 x^{\frac{4}{5}} x + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.4

Raise $x$ to the power of $1$ .

$d \int - 9 (x^{\frac{4}{5}} x^{1}) + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.5

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

$d \int - 9 x^{\frac{4}{5} + 1} + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.6

Write $1$ as a fraction with a common denominator.

$d \int - 9 x^{\frac{4}{5} + \frac{5}{5}} + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.7

Combine the numerators over the common denominator.

$d \int - 9 x^{\frac{4 + 5}{5}} + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.8

Add $4$ and $5$ .

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 6} x + 9 x^{3} x - 15 x d x$

Step 5.9

Raise $x$ to the power of $1$ .

$d \int - 9 x^{\frac{9}{5}} + 3 (x^{- 6} x^{1}) + 9 x^{3} x - 15 x d x$

Step 5.10

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

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 6 + 1} + 9 x^{3} x - 15 x d x$

Step 5.11

Add $- 6$ and $1$ .

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 5} + 9 x^{3} x - 15 x d x$

Step 5.12

Raise $x$ to the power of $1$ .

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 5} + 9 (x^{3} x^{1}) - 15 x d x$

Step 5.13

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

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 5} + 9 x^{3 + 1} - 15 x d x$

Step 5.14

Add $3$ and $1$ .

$d \int - 9 x^{\frac{9}{5}} + 3 x^{- 5} + 9 x^{4} - 15 x d x$

Step 5.15

Reorder $- 9 x^{\frac{9}{5}}$ and $3 x^{- 5}$ .

$d \int 3 x^{- 5} - 9 x^{\frac{9}{5}} + 9 x^{4} - 15 x d x$

Step 5.16

Move $- 9 x^{\frac{9}{5}}$ .

$d \int 3 x^{- 5} + 9 x^{4} - 9 x^{\frac{9}{5}} - 15 x d x$

Step 5.17

Reorder $3 x^{- 5}$ and $9 x^{4}$ .

$d \int 9 x^{4} + 3 x^{- 5} - 9 x^{\frac{9}{5}} - 15 x d x$

Step 5.18

Move $- 9 x^{\frac{9}{5}}$ .

$d \int 9 x^{4} + 3 x^{- 5} - 15 x - 9 x^{\frac{9}{5}} d x$

Step 5.19

Move $3 x^{- 5}$ .

$d \int 9 x^{4} - 15 x + 3 x^{- 5} - 9 x^{\frac{9}{5}} d x$

Step 6

Split the single integral into multiple integrals.

$d (\int 9 x^{4} d x + \int - 15 x d x + \int 3 x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 7

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$d (9 \int x^{4} d x + \int - 15 x d x + \int 3 x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 8

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

$d (9 (\frac{1}{5} x^{5} + C) + \int - 15 x d x + \int 3 x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 9

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

$d (9 (\frac{1}{5} x^{5} + C) - 15 \int x d x + \int 3 x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 10

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

$d (9 (\frac{1}{5} x^{5} + C) - 15 (\frac{1}{2} x^{2} + C) + \int 3 x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 11

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$d (9 (\frac{1}{5} x^{5} + C) - 15 (\frac{1}{2} x^{2} + C) + 3 \int x^{- 5} d x + \int - 9 x^{\frac{9}{5}} d x)$

Step 12

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

$d (9 (\frac{1}{5} x^{5} + C) - 15 (\frac{1}{2} x^{2} + C) + 3 (- \frac{1}{4} x^{- 4} + C) + \int - 9 x^{\frac{9}{5}} d x)$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{1}{2} x^{2} + C) + 3 (- \frac{1}{4} x^{- 4} + C) + \int - 9 x^{\frac{9}{5}} d x)$

Step 13.2

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

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{4} x^{- 4} + C) + \int - 9 x^{\frac{9}{5}} d x)$

Step 13.3

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

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{x^{- 4}}{4} + C) + \int - 9 x^{\frac{9}{5}} d x)$

Step 13.4

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

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{4 x^{4}} + C) + \int - 9 x^{\frac{9}{5}} d x)$

Step 14

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{4 x^{4}} + C) - 9 \int x^{\frac{9}{5}} d x)$

Step 15

By the Power Rule, the integral of $x^{\frac{9}{5}}$ with respect to $x$ is $\frac{5}{14} x^{\frac{14}{5}}$ .

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{4 x^{4}} + C) - 9 (\frac{5}{14} x^{\frac{14}{5}} + C))$

Step 16

Simplify.

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Step 16.1

Combine $\frac{5}{14}$ and $x^{\frac{14}{5}}$ .

$d (9 (\frac{x^{5}}{5} + C) - 15 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{4 x^{4}} + C) - 9 (\frac{5 x^{\frac{14}{5}}}{14} + C))$

Step 16.2

Simplify.

$d (\frac{9 x^{5}}{5} - \frac{15 x^{2}}{2} - \frac{3}{4 x^{4}} - \frac{45 x^{\frac{14}{5}}}{14}) + C$

Step 17

Reorder terms.

$d (\frac{9}{5} x^{5} - \frac{15}{2} x^{2} - \frac{3}{4 x^{4}} - \frac{45}{14} x^{\frac{14}{5}}) + C$

Step 18

The answer is the antiderivative of the function $f (x) = (- 9 x^{\frac{4}{5}} + 3 x^{- 6} + 9 x^{3} - 15) d x$ .

$F (x) =$ $d (\frac{9}{5} x^{5} - \frac{15}{2} x^{2} - \frac{3}{4 x^{4}} - \frac{45}{14} x^{\frac{14}{5}}) + C$