Calculus Examples

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

Step 1

Since $\frac{x}{1 + x^{2}}$ is constant with respect to $d$ , move $\frac{x}{1 + x^{2}}$ out of the integral.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

$\frac{x}{1 + x^{2}} \cdot \frac{1}{2} d^{2} + C$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{x}{(1 + x^{2}) \cdot 2} d^{2} + C$

Step 3.2.2

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

$\frac{x}{2 \cdot (1 + x^{2})} d^{2} + C$

Step 3.2.3

Combine $\frac{x}{2 (1 + x^{2})}$ and $d^{2}$ .

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