Calculus Examples

$f (x) = \frac{2 + x^{2}}{1 + x^{2}}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

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

Step 3

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

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

Step 4

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

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

Step 5

Divide $x^{2} + 2$ by $x^{2} + 1$ .

Tap for more steps...

Step 5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$2$

Step 5.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x^{2}$ .

							$1$
	$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$2$

Step 5.3

Multiply the new quotient term by the divisor.

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$2$
					+	$x^{2}$	+	$0$	+	$1$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 + 1$

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$2$
					-	$x^{2}$	-	$0$	-	$1$

Step 5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$2$
					-	$x^{2}$	-	$0$	-	$1$
									+	$1$

Step 5.6

The final answer is the quotient plus the remainder over the divisor.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 9

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

$x + C + \arctan (x) + C$

Step 10

Simplify.

$x + \arctan (x) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{2 + x^{2}}{1 + x^{2}}$ .

$F (x) =$ $x + \arctan (x) + C$