Calculus Examples

$\int \frac{- 27 x^{2}}{{(9 - 9 x^{3})}^{2}} d x$

Step 1

Move the negative in front of the fraction.

$\int - \frac{27 x^{2}}{{(9 - 9 x^{3})}^{2}} d x$

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{27 x^{2}}{{(9 - 9 x^{3})}^{2}} d x$

Step 3

Since $27$ is constant with respect to $x$ , move $27$ out of the integral.

$- (27 \int \frac{x^{2}}{{(9 - 9 x^{3})}^{2}} d x)$

Step 4

Multiply $27$ by $- 1$ .

$- 27 \int \frac{x^{2}}{{(9 - 9 x^{3})}^{2}} d x$

Step 5

Let $u = 9 - 9 x^{3}$ . Then $d u = - 27 x^{2} d x$ , so $- \frac{1}{27} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 9 - 9 x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $9 - 9 x^{3}$ .

$\frac{d}{d x} [9 - 9 x^{3}]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $9 - 9 x^{3}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x^{3}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x^{3}]$

Step 5.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 9 x^{3}]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 9 x^{3}]$ .

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Step 5.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{3}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{3}]$ .

$0 - 9 \frac{d}{d x} [x^{3}]$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 - 9 (3 x^{2})$

Step 5.1.3.3

Multiply $3$ by $- 9$ .

$0 - 27 x^{2}$

Step 5.1.4

Subtract $27 x^{2}$ from $0$ .

$- 27 x^{2}$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$- 27 \int \frac{1}{u^{2}} \cdot \frac{1}{- 27} d u$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

$- 27 \int \frac{1}{u^{2}} (- \frac{1}{27}) d u$

Step 6.2

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{27}$ .

$- 27 \int - \frac{1}{u^{2} \cdot 27} d u$

Step 6.3

Move $27$ to the left of $u^{2}$ .

$- 27 \int - \frac{1}{27 u^{2}} d u$

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- 27 (- \int \frac{1}{27 u^{2}} d u)$

Step 8

Multiply $- 1$ by $- 27$ .

$27 \int \frac{1}{27 u^{2}} d u$

Step 9

Since $\frac{1}{27}$ is constant with respect to $u$ , move $\frac{1}{27}$ out of the integral.

$27 (\frac{1}{27} \int \frac{1}{u^{2}} d u)$

Step 10

Simplify the expression.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{27}$ and $27$ .

$\frac{27}{27} \int \frac{1}{u^{2}} d u$

Step 10.1.2

Cancel the common factor of $27$ .

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Step 10.1.2.1

Cancel the common factor.

$\frac{27}{27} \int \frac{1}{u^{2}} d u$

Step 10.1.2.2

Rewrite the expression.

$1 \int \frac{1}{u^{2}} d u$

Step 10.1.3

Multiply $\int \frac{1}{u^{2}} d u$ by $1$ .

$\int \frac{1}{u^{2}} d u$

Step 10.2

Apply basic rules of exponents.

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Step 10.2.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{2})}^{- 1} d u$

Step 10.2.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 10.2.2.1

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

$\int u^{2 \cdot - 1} d u$

Step 10.2.2.2

Multiply $2$ by $- 1$ .

$\int u^{- 2} d u$

Step 11

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$- u^{- 1} + C$

Step 12

Rewrite $- u^{- 1} + C$ as $- \frac{1}{u} + C$ .

$- \frac{1}{u} + C$

Step 13

Replace all occurrences of $u$ with $9 - 9 x^{3}$ .

$- \frac{1}{9 - 9 x^{3}} + C$