Calculus Examples

$\int \frac{9 x^{2}}{{(7 - 3 x^{3})}^{\frac{1}{2}}} d x$

Step 1

Let $u = 7 - 3 x^{3}$ . Then $d u = - 9 x^{2} d x$ , so $\frac{1}{- 9 x^{2}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 7 - 3 x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $7 - 3 x^{3}$ .

$\frac{d}{d x} [7 - 3 x^{3}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $7 - 3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x^{3}]$ .

$\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x^{3}]$

Step 1.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $3$ by $- 3$ .

$0 - 9 x^{2}$

Step 1.1.4

Subtract $9 x^{2}$ from $0$ .

$- 9 x^{2}$

Step 1.2

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 5

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

$- (2 u^{\frac{1}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $- (2 u^{\frac{1}{2}} + C)$ as $- 1 \cdot 2 u^{\frac{1}{2}} + C$ .

$- 1 \cdot 2 u^{\frac{1}{2}} + C$

Step 6.2

Rewrite $- 1 \cdot 2 u^{\frac{1}{2}} + C$ as $- 2 u^{\frac{1}{2}} + C$ .

$- 2 u^{\frac{1}{2}} + C$

Step 7

Replace all occurrences of $u$ with $7 - 3 x^{3}$ .

$- 2 {(7 - 3 x^{3})}^{\frac{1}{2}} + C$