Calculus Examples

$\int \frac{x}{x^{4} + 2 x^{2} + 1} d x$

Step 1

Write the fraction using partial fraction decomposition.

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Step 1.1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1.1

Factor the fraction.

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Step 1.1.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.1.1.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 1.1.1.3

Factor using the perfect square rule.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

$2 u = 2 \cdot u \cdot 1$

Step 1.1.1.3.3

Rewrite the polynomial.

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

Step 1.1.1.3.4

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

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

Step 1.1.1.4

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.1.2

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 is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

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

Step 1.1.3

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

Step 1.1.4

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

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

Step 1.1.5

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

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Step 1.1.5.1

Cancel the common factor.

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

Step 1.1.5.2

Divide $x$ by $1$ .

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

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

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

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Step 1.1.6.1.1

Cancel the common factor.

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

Step 1.1.6.1.2

Divide $A x + B$ by $1$ .

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

Step 1.1.6.2

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

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Step 1.1.6.2.1

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

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

Step 1.1.6.2.2

Cancel the common factors.

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Step 1.1.6.2.2.1

Multiply by $1$ .

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

Step 1.1.6.2.2.2

Cancel the common factor.

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

Step 1.1.6.2.2.3

Rewrite the expression.

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

Step 1.1.6.2.2.4

Divide $(C x + D) (x^{2} + 1)$ by $1$ .

$x = A x + B + (C x + D) (x^{2} + 1)$

Step 1.1.6.3

Expand $(C x + D) (x^{2} + 1)$ using the FOIL Method.

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Step 1.1.6.3.1

Apply the distributive property.

$x = A x + B + C x (x^{2} + 1) + D (x^{2} + 1)$

Step 1.1.6.3.2

Apply the distributive property.

$x = A x + B + C x \cdot x^{2} + C x \cdot 1 + D (x^{2} + 1)$

Step 1.1.6.3.3

Apply the distributive property.

$x = A x + B + C x \cdot x^{2} + C x \cdot 1 + D x^{2} + D \cdot 1$

Step 1.1.6.4

Simplify each term.

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Step 1.1.6.4.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.1.6.4.1.1

Move $x^{2}$ .

$x = A x + B + C (x^{2} x) + C x \cdot 1 + D x^{2} + D \cdot 1$

Step 1.1.6.4.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.1.6.4.1.2.1

Raise $x$ to the power of $1$ .

$x = A x + B + C (x^{2} x) + C x \cdot 1 + D x^{2} + D \cdot 1$

Step 1.1.6.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = A x + B + C x^{2 + 1} + C x \cdot 1 + D x^{2} + D \cdot 1$

Step 1.1.6.4.1.3

Add $2$ and $1$ .

$x = A x + B + C x^{3} + C x \cdot 1 + D x^{2} + D \cdot 1$

Step 1.1.6.4.2

Multiply $C$ by $1$ .

$x = A x + B + C x^{3} + C x + D x^{2} + D \cdot 1$

Step 1.1.6.4.3

Multiply $D$ by $1$ .

$x = A x + B + C x^{3} + C x + D x^{2} + D$

Step 1.1.7

Simplify the expression.

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Step 1.1.7.1

Move $B$ .

$x = A x + C x^{3} + C x + D x^{2} + B + D$

Step 1.1.7.2

Move $C x$ .

$x = A x + C x^{3} + D x^{2} + C x + B + D$

Step 1.1.7.3

Move $A x$ .

$x = C x^{3} + D x^{2} + A x + C x + B + D$

Step 1.2

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

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Step 1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = C$

Step 1.2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = D$

Step 1.2.3

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.

$1 = A + C$

Step 1.2.4

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.

$0 = B + D$

Step 1.2.5

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

$0 = C$

$0 = D$

$1 = A + C$

$0 = B + D$

$0 = C$

$0 = D$

$1 = A + C$

$0 = B + D$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$0 = D$

$1 = A + C$

$0 = B + D$

Step 1.3.2

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

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Step 1.3.2.1

Rewrite the equation as $D = 0$ .

$D = 0$

$C = 0$

$1 = A + C$

$0 = B + D$

Step 1.3.2.2

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

$1 = A + 0$

$D = 0$

$C = 0$

$0 = B + D$

Step 1.3.2.3

Simplify $1 = A + 0$ .

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Step 1.3.2.3.1

Simplify the left side.

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Step 1.3.2.3.1.1

Remove parentheses.

$1 = A + 0$

$D = 0$

$C = 0$

$0 = B + D$

$1 = A + 0$

$D = 0$

$C = 0$

$0 = B + D$

Step 1.3.2.3.2

Simplify the right side.

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Step 1.3.2.3.2.1

Add $A$ and $0$ .

$1 = A$

$D = 0$

$C = 0$

$0 = B + D$

$1 = A$

$D = 0$

$C = 0$

$0 = B + D$

$1 = A$

$D = 0$

$C = 0$

$0 = B + D$

$1 = A$

$D = 0$

$C = 0$

$0 = B + D$

Step 1.3.3

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

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Step 1.3.3.1

Rewrite the equation as $A = 1$ .

$A = 1$

$D = 0$

$C = 0$

$0 = B + D$

Step 1.3.3.2

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

$0 = B + 0$

$A = 1$

$D = 0$

$C = 0$

Step 1.3.3.3

Simplify $0 = B + 0$ .

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Step 1.3.3.3.1

Simplify the left side.

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Step 1.3.3.3.1.1

Remove parentheses.

$0 = B + 0$

$A = 1$

$D = 0$

$C = 0$

$0 = B + 0$

$A = 1$

$D = 0$

$C = 0$

Step 1.3.3.3.2

Simplify the right side.

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Step 1.3.3.3.2.1

Add $B$ and $0$ .

$0 = B$

$A = 1$

$D = 0$

$C = 0$

$0 = B$

$A = 1$

$D = 0$

$C = 0$

$0 = B$

$A = 1$

$D = 0$

$C = 0$

$0 = B$

$A = 1$

$D = 0$

$C = 0$

Step 1.3.4

Rewrite the equation as $B = 0$ .

$B = 0$

$A = 1$

$D = 0$

$C = 0$

Step 1.3.5

Solve the system of equations.

$B = 0 A = 1 D = 0 C = 0$

Step 1.3.6

List all of the solutions.

$B = 0, A = 1, D = 0, C = 0$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Multiply $x$ by $1$ .

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

Step 1.5.1.2

Add $x$ and $0$ .

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

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

Multiply $0$ by $x$ .

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

Step 1.5.2.2

Add $0$ and $0$ .

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

Step 1.5.3

Divide $0$ by $x^{2} + 1$ .

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

Step 1.5.4

Remove the zero from the expression.

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

Step 2

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 2.1.4

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

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$\int \frac{1}{u^{2}} \cdot \frac{1}{2} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{2}$ .

$\int \frac{1}{u^{2} \cdot 2} d u$

Step 3.2

Move $2$ to the left of $u^{2}$ .

$\int \frac{1}{2 u^{2}} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{1}{u^{2}} d u$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{1}{2} (- u^{- 1} + C)$ as $\frac{1}{2} (- \frac{1}{u}) + C$ .

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

Step 7.2

Multiply $\frac{1}{2}$ by $\frac{1}{u}$ .

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

Step 8

Replace all occurrences of $u$ with $x^{2} + 1$ .

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