Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 4

Apply the constant rule.

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

Step 5

Simplify the expression.

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Step 5.1

Reorder $x^{2}$ and $1$ .

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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