Calculus Examples

$5 - 3 {(1 + x^{2})}^{- 1}$

Step 1

Write $5 - 3 {(1 + x^{2})}^{- 1}$ as a function.

$f (x) = 5 - 3 {(1 + x^{2})}^{- 1}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 5 - 3 {(1 + x^{2})}^{- 1} d x$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int 5 - 3 \frac{1}{1 + x^{2}} d x$

Step 4.1.2

Combine $- 3$ and $\frac{1}{1 + x^{2}}$ .

$\int 5 + \frac{- 3}{1 + x^{2}} d x$

Step 4.1.3

Move the negative in front of the fraction.

$\int 5 - \frac{3}{1 + x^{2}} d x$

Step 4.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

$\int \frac{5 (1 + x^{2})}{1 + x^{2}} - \frac{3}{1 + x^{2}} d x$

Step 4.3

Combine the numerators over the common denominator.

$\int \frac{5 (1 + x^{2}) - 3}{1 + x^{2}} d x$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Apply the distributive property.

$\int \frac{5 \cdot 1 + 5 x^{2} - 3}{1 + x^{2}} d x$

Step 4.4.2

Multiply $5$ by $1$ .

$\int \frac{5 + 5 x^{2} - 3}{1 + x^{2}} d x$

Step 4.4.3

Subtract $3$ from $5$ .

$\int \frac{5 x^{2} + 2}{1 + x^{2}} d x$

Step 5

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

$\int \frac{5 x^{2} + 2}{x^{2} + 1} d x$

Step 6

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

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Step 6.1

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

$x^{2}$

$0 x$

$1$

$5 x^{2}$

$0 x$

$2$

Step 6.2

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x^{2}$ .

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

Step 6.3

Multiply the new quotient term by the divisor.

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

Step 6.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 0 + 5$

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

Step 6.5

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

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

Step 6.6

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

$\int 5 - \frac{3}{x^{2} + 1} d x$

Step 7

Split the single integral into multiple integrals.

$\int 5 d x + \int - \frac{3}{x^{2} + 1} d x$

Step 8

Apply the constant rule.

$5 x + C + \int - \frac{3}{x^{2} + 1} d x$

Step 9

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$5 x + C - \int \frac{3}{x^{2} + 1} d x$

Step 10

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

$5 x + C - (3 \int \frac{1}{x^{2} + 1} d x)$

Step 11

Simplify the expression.

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Step 11.1

Multiply $3$ by $- 1$ .

$5 x + C - 3 \int \frac{1}{x^{2} + 1} d x$

Step 11.2

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

$5 x + C - 3 \int \frac{1}{1 + x^{2}} d x$

Step 11.3

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

$5 x + C - 3 \int \frac{1}{1^{2} + x^{2}} d x$

Step 12

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

$5 x + C - 3 (\arctan (x) + C)$

Step 13

Simplify.

$5 x - 3 \arctan (x) + C$

Step 14

The answer is the antiderivative of the function $f (x) = 5 - 3 {(1 + x^{2})}^{- 1}$ .

$F (x) =$ $5 x - 3 \arctan (x) + C$