Calculus Examples

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

Step 1

Write $\frac{x^{3} + 3 x^{2} + 3 x - 1}{x^{2} + 2 x + 1}$ as a function.

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

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.

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

Step 4

Divide $x^{3} + 3 x^{2} + 3 x - 1$ by $x^{2} + 2 x + 1$ .

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Step 4.1

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

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 4.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 4.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

Step 4.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 2 x^{2} + x$

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

Step 4.5

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

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

$x^{2}$

$2 x$

Step 4.6

Pull the next terms from the original dividend down into the current dividend.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

Step 4.7

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

$x$

$1$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

Step 4.8

Multiply the new quotient term by the divisor.

$x$

$1$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{2}$

$2 x$

$1$

Step 4.9

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

$x$

$1$

$x^{2}$

$2 x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$x^{3}$

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{2}$

$2 x$

$1$

Step 4.10

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

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

Step 4.11

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

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

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

Step 9

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 10

Multiply $2$ by $- 1$ .

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

Step 11

Write the fraction using partial fraction decomposition.

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Step 11.1

Decompose the fraction and multiply through by the common denominator.

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Step 11.1.1

Factor using the perfect square rule.

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Step 11.1.1.1

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

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

Step 11.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 11.1.1.3

Rewrite the polynomial.

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

Step 11.1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 11.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

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

Step 11.1.3

$\frac{A}{{(x + 1)}^{2}} + \frac{B}{x + 1}$

Step 11.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x + 1)}^{2}$ .

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

Step 11.1.5

Cancel the common factor of ${(x + 1)}^{2}$ .

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Step 11.1.5.1

Cancel the common factor.

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

Step 11.1.5.2

Rewrite the expression.

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

Step 11.1.6

Simplify each term.

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Step 11.1.6.1

Cancel the common factor of ${(x + 1)}^{2}$ .

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Step 11.1.6.1.1

Cancel the common factor.

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

Step 11.1.6.1.2

Divide $A$ by $1$ .

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

Step 11.1.6.2

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 11.1.6.2.1

Factor $x + 1$ out of $(B) {(x + 1)}^{2}$ .

$1 = A + \frac{(x + 1) (B (x + 1))}{x + 1}$

Step 11.1.6.2.2

Cancel the common factors.

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Step 11.1.6.2.2.1

Multiply by $1$ .

$1 = A + \frac{(x + 1) (B (x + 1))}{(x + 1) \cdot 1}$

Step 11.1.6.2.2.2

Cancel the common factor.

$1 = A + \frac{(x + 1) (B (x + 1))}{(x + 1) \cdot 1}$

Step 11.1.6.2.2.3

Rewrite the expression.

$1 = A + \frac{B (x + 1)}{1}$

Step 11.1.6.2.2.4

Divide $B (x + 1)$ by $1$ .

$1 = A + B (x + 1)$

Step 11.1.6.3

Apply the distributive property.

$1 = A + B x + B \cdot 1$

Step 11.1.6.4

Multiply $B$ by $1$ .

$1 = A + B x + B$

Step 11.1.7

Reorder $A$ and $B x$ .

$1 = B x + A + B$

Step 11.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 11.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = B$

Step 11.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B$

Step 11.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = B$

$1 = A + B$

$0 = B$

$1 = A + B$

Step 11.3

Solve the system of equations.

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Step 11.3.1

Rewrite the equation as $B = 0$ .

$B = 0$

$1 = A + B$

Step 11.3.2

Replace all occurrences of $B$ with $0$ in each equation.

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Step 11.3.2.1

Replace all occurrences of $B$ in $1 = A + B$ with $0$ .

$1 = A + 0$

$B = 0$

Step 11.3.2.2

Simplify $1 = A + 0$ .

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Step 11.3.2.2.1

Simplify the left side.

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Step 11.3.2.2.1.1

Remove parentheses.

$1 = A + 0$

$B = 0$

$1 = A + 0$

$B = 0$

Step 11.3.2.2.2

Simplify the right side.

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Step 11.3.2.2.2.1

Add $A$ and $0$ .

$1 = A$

$B = 0$

$1 = A$

$B = 0$

$1 = A$

$B = 0$

$1 = A$

$B = 0$

Step 11.3.3

Rewrite the equation as $A = 1$ .

$A = 1$

$B = 0$

Step 11.3.4

Solve the system of equations.

$A = 1 B = 0$

Step 11.3.5

List all of the solutions.

$A = 1, B = 0$

Step 11.4

Replace each of the partial fraction coefficients in $\frac{A}{{(x + 1)}^{2}} + \frac{B}{x + 1}$ with the values found for $A$ and $B$ .

$\frac{1}{{(x + 1)}^{2}} + \frac{0}{x + 1}$

Step 11.5

Simplify.

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Step 11.5.1

Divide $0$ by $x + 1$ .

$\frac{1}{{(x + 1)}^{2}} + 0$

Step 11.5.2

Remove the zero from the expression.

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

Step 12

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 12.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 12.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 12.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 12.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 12.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 12.1.5

Add $1$ and $0$ .

$1$

Step 12.2

Rewrite the problem using $u$ and $d u$ .

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

Step 13

Apply basic rules of exponents.

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Step 13.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 13.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 13.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 13.2.2

Multiply $2$ by $- 1$ .

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

Step 14

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

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

Step 15

Simplify.

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Step 15.1

Simplify.

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

Step 15.2

Multiply $- 1$ by $- 2$ .

$\frac{1}{2} x^{2} + x + 2 u^{- 1} + C$

Step 16

Replace all occurrences of $u$ with $x + 1$ .

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

Step 17

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

$F (x) =$ $\frac{1}{2} x^{2} + x + 2 {(x + 1)}^{- 1} + C$