Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $x^{3}$ .

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

Step 3.1.2

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 3.2

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

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

Step 4

Simplify.

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Step 4.1

Simplify.

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $3$ .

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

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

Multiply $\int \frac{1}{1 + u_{1}} d u_{1}$ by $1$ .

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $1 + u_{1}$ .

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

Step 7.1.2

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

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

Step 7.1.3

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

$0 + \frac{d}{d u_{1}} [u_{1}]$

Step 7.1.4

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

$0 + 1$

Step 7.1.5

Add $0$ and $1$ .

$1$

Step 7.2

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

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

Step 8

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

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

Step 9

Substitute back in for each integration substitution variable.

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Step 9.1

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

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

Step 9.2

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

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