Calculus Examples

$\frac{x^{2}}{x^{3} + 1}$

Step 1

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

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

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 $d u_{1}$ .

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

Step 3

Simplify.

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Step 3.1

Simplify.

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Let $u_{2} = u_{1} + 1$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 5.1

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

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Step 5.1.1

Differentiate $u_{1} + 1$ .

$\frac{d}{d u_{1}} [u_{1} + 1]$

Step 5.1.2

By the Sum Rule, the derivative of $u_{1} + 1$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [1]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [1]$

Step 5.1.3

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

$1 + \frac{d}{d u_{1}} [1]$

Step 5.1.4

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

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

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

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

Step 7

Simplify.

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

Step 8

Substitute back in for each integration substitution variable.

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Step 8.1

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

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

Step 8.2

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

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