Calculus Examples

$f (x) = \frac{2 x^{2} + 5}{x^{2} + 1}$

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.

$F (x) = \int \frac{2 x^{2} + 5}{x^{2} + 1} d x$

Step 3

Divide $2 x^{2} + 5$ by $x^{2} + 1$ .

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Step 3.1

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

$x^{2}$

$0 x$

$1$

$2 x^{2}$

$0 x$

$5$

Step 3.2

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

							$2$
	$x^{2}$	+	$0 x$	+	$1$		$2 x^{2}$	+	$0 x$	+	$5$

Step 3.3

Multiply the new quotient term by the divisor.

						$2$
$x^{2}$	+	$0 x$	+	$1$		$2 x^{2}$	+	$0 x$	+	$5$
					+	$2 x^{2}$	+	$0$	+	$2$

Step 3.4

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

						$2$
$x^{2}$	+	$0 x$	+	$1$		$2 x^{2}$	+	$0 x$	+	$5$
					-	$2 x^{2}$	-	$0$	-	$2$

Step 3.5

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

						$2$
$x^{2}$	+	$0 x$	+	$1$		$2 x^{2}$	+	$0 x$	+	$5$
					-	$2 x^{2}$	-	$0$	-	$2$
									+	$3$

Step 3.6

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

$2 x + C + 3 (\arctan (x) + C)$

Step 9

Simplify.

$2 x + 3 \arctan (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{2 x^{2} + 5}{x^{2} + 1}$ .

$F (x) =$ $2 x + 3 \arctan (x) + C$