Calculus Examples

$\int \frac{x^{2}}{3 - x^{3}} d x$

Step 1

Rewrite $x^{3}$ as ${(x^{3})}^{1}$ .

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

Step 2

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

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Step 2.1

Let $u_{1} = x^{3}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $x^{3}$ .

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

Step 2.1.2

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

$3 x^{2}$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{3 - {u_{1}}^{1}} \cdot \frac{1}{3} d u_{1}$

Step 3

Simplify.

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Step 3.1

Simplify.

$\int \frac{1}{3 - u_{1}} \cdot \frac{1}{3} d u_{1}$

Step 3.2

Multiply $\frac{1}{3 - u_{1}}$ by $\frac{1}{3}$ .

$\int \frac{1}{(3 - u_{1}) \cdot 3} d u_{1}$

Step 3.3

Move $3$ to the left of $3 - u_{1}$ .

$\int \frac{1}{3 (3 - u_{1})} d u_{1}$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{3 - u_{1}} d u_{1}$

Step 5

Let $u_{2} = 3 - u_{1}$ . Then $d u_{2} = - d u_{1}$ , so $- d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 5.1

Let $u_{2} = 3 - u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Rewrite.

$\frac{- 1}{1}$

Step 5.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{3} \int \frac{- 1}{u_{2}} d u_{2}$

Step 6

Move the negative in front of the fraction.

$\frac{1}{3} \int - \frac{1}{u_{2}} d u_{2}$

Step 7

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\frac{1}{3} (- \int \frac{1}{u_{2}} d u_{2})$

Step 8

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{3} (- (\ln (| u_{2} |) + C))$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{1}{3} (- \ln (| u_{2} |)) + C$

Step 9.2

Combine $\frac{1}{3}$ and $\ln (| u_{2} |)$ .

$- \frac{\ln (| u_{2} |)}{3} + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

Replace all occurrences of $u_{2}$ with $3 - u_{1}$ .

$- \frac{\ln (| 3 - u_{1} |)}{3} + C$

Step 10.2

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$- \frac{\ln (| 3 - x^{3} |)}{3} + C$

Step 11

Reorder terms.

$- \frac{1}{3} \ln (| 3 - x^{3} |) + C$