Calculus Examples

$\int \frac{x^{9}}{1 + x^{20}} d x$

Step 1

Rewrite $x^{20}$ as ${(x^{10})}^{2}$ .

$\int \frac{x^{9}}{1 + {(x^{10})}^{2}} d x$

Step 2

Let $u = x^{10}$ . Then $d u = 10 x^{9} d x$ , so $\frac{1}{10} d u = x^{9} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x^{10}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{10}$ .

$\frac{d}{d x} [x^{10}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$10 x^{9}$

Step 2.2

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

$\int \frac{1}{1 + u^{2}} \cdot \frac{1}{10} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{1 + u^{2}}$ by $\frac{1}{10}$ .

$\int \frac{1}{(1 + u^{2}) \cdot 10} d u$

Step 3.2

Move $10$ to the left of $1 + u^{2}$ .

$\int \frac{1}{10 (1 + u^{2})} d u$

Step 4

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$\frac{1}{10} \int \frac{1}{1 + u^{2}} d u$

Step 5

Rewrite $1$ as $1^{2}$ .

$\frac{1}{10} \int \frac{1}{1^{2} + u^{2}} d u$

Step 6

The integral of $\frac{1}{1^{2} + u^{2}}$ with respect to $u$ is $\arctan (u) + C$ .

$\frac{1}{10} (\arctan (u) + C)$

Step 7

Simplify.

$\frac{1}{10} \arctan (u) + C$

Step 8

Replace all occurrences of $u$ with $x^{10}$ .

$\frac{1}{10} \arctan (x^{10}) + C$