Calculus Examples

$\int \frac{3 u - 3}{{(u^{2} - 2 u + 6)}^{2}} d u$

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 $3$ out of $3 u - 3$ .

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Step 1.1.1.1

Factor $3$ out of $3 u$ .

$\frac{3 (u) - 3}{{(u^{2} - 2 u + 6)}^{2}}$

Step 1.1.1.2

Factor $3$ out of $- 3$ .

$\frac{3 (u) + 3 (- 1)}{{(u^{2} - 2 u + 6)}^{2}}$

Step 1.1.1.3

Factor $3$ out of $3 (u) + 3 (- 1)$ .

$\frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}}$

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.

$\frac{A u + B}{{(u^{2} - 2 u + 6)}^{2}}$

Step 1.1.3

$\frac{A u + B}{{(u^{2} - 2 u + 6)}^{2}} + \frac{C u + D}{u^{2} - 2 u + 6}$

Step 1.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(u^{2} - 2 u + 6)}^{2}$ .

$\frac{3 (u - 1) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.5

Cancel the common factor of ${(u^{2} - 2 u + 6)}^{2}$ .

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Step 1.1.5.1

Cancel the common factor.

$\frac{3 (u - 1) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.5.2

Divide $3 (u - 1)$ by $1$ .

$3 (u - 1) = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.6

Apply the distributive property.

$3 u + 3 \cdot - 1 = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.7

Multiply $3$ by $- 1$ .

$3 u - 3 = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.8

Simplify each term.

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Step 1.1.8.1

Cancel the common factor of ${(u^{2} - 2 u + 6)}^{2}$ .

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Step 1.1.8.1.1

Cancel the common factor.

$3 u - 3 = \frac{(A u + B) {(u^{2} - 2 u + 6)}^{2}}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.8.1.2

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

$3 u - 3 = A u + B + \frac{(C u + D) {(u^{2} - 2 u + 6)}^{2}}{u^{2} - 2 u + 6}$

Step 1.1.8.2

Cancel the common factor of ${(u^{2} - 2 u + 6)}^{2}$ and $u^{2} - 2 u + 6$ .

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Step 1.1.8.2.1

Factor $u^{2} - 2 u + 6$ out of $(C u + D) {(u^{2} - 2 u + 6)}^{2}$ .

$3 u - 3 = A u + B + \frac{(u^{2} - 2 u + 6) ((C u + D) (u^{2} - 2 u + 6))}{u^{2} - 2 u + 6}$

Step 1.1.8.2.2

Cancel the common factors.

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Step 1.1.8.2.2.1

Multiply by $1$ .

$3 u - 3 = A u + B + \frac{(u^{2} - 2 u + 6) ((C u + D) (u^{2} - 2 u + 6))}{(u^{2} - 2 u + 6) \cdot 1}$

Step 1.1.8.2.2.2

Cancel the common factor.

$3 u - 3 = A u + B + \frac{(u^{2} - 2 u + 6) ((C u + D) (u^{2} - 2 u + 6))}{(u^{2} - 2 u + 6) \cdot 1}$

Step 1.1.8.2.2.3

Rewrite the expression.

$3 u - 3 = A u + B + \frac{(C u + D) (u^{2} - 2 u + 6)}{1}$

Step 1.1.8.2.2.4

Divide $(C u + D) (u^{2} - 2 u + 6)$ by $1$ .

$3 u - 3 = A u + B + (C u + D) (u^{2} - 2 u + 6)$

Step 1.1.8.3

Expand $(C u + D) (u^{2} - 2 u + 6)$ by multiplying each term in the first expression by each term in the second expression.

$3 u - 3 = A u + B + C u \cdot u^{2} + C u (- 2 u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4

Simplify each term.

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Step 1.1.8.4.1

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

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Step 1.1.8.4.1.1

Move $u^{2}$ .

$3 u - 3 = A u + B + C (u^{2} u) + C u (- 2 u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.1.2

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

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Step 1.1.8.4.1.2.1

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

$3 u - 3 = A u + B + C (u^{2} u) + C u (- 2 u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.1.2.2

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

$3 u - 3 = A u + B + C u^{2 + 1} + C u (- 2 u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.1.3

Add $2$ and $1$ .

$3 u - 3 = A u + B + C u^{3} + C u (- 2 u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.2

Rewrite using the commutative property of multiplication.

$3 u - 3 = A u + B + C u^{3} - 2 C u \cdot u + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.3

Multiply $u$ by $u$ by adding the exponents.

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Step 1.1.8.4.3.1

Move $u$ .

$3 u - 3 = A u + B + C u^{3} - 2 C (u \cdot u) + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.3.2

Multiply $u$ by $u$ .

$3 u - 3 = A u + B + C u^{3} - 2 C u^{2} + C u \cdot 6 + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.4

Move $6$ to the left of $C u$ .

$3 u - 3 = A u + B + C u^{3} - 2 C u^{2} + 6 \cdot (C u) + D u^{2} + D (- 2 u) + D \cdot 6$

Step 1.1.8.4.5

Rewrite using the commutative property of multiplication.

$3 u - 3 = A u + B + C u^{3} - 2 C u^{2} + 6 C u + D u^{2} - 2 D u + D \cdot 6$

Step 1.1.8.4.6

Move $6$ to the left of $D$ .

$3 u - 3 = A u + B + C u^{3} - 2 C u^{2} + 6 C u + D u^{2} - 2 D u + 6 D$

Step 1.1.9

Simplify the expression.

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Step 1.1.9.1

Move $B$ .

$3 u - 3 = A u + C u^{3} - 2 C u^{2} + 6 C u + D u^{2} - 2 D u + B + 6 D$

Step 1.1.9.2

Move $6 C u$ .

$3 u - 3 = A u + C u^{3} - 2 C u^{2} + D u^{2} + 6 C u - 2 D u + B + 6 D$

Step 1.1.9.3

Move $A u$ .

$3 u - 3 = C u^{3} - 2 C u^{2} + D u^{2} + A u + 6 C u - 2 D u + B + 6 D$

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 $u^{3}$ 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 $u^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 2 C + D$

Step 1.2.3

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

$3 = A + 6 C - 2 D$

Step 1.2.4

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

$- 3 = B + 6 D$

Step 1.2.5

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

$0 = C$

$0 = - 2 C + D$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

$0 = C$

$0 = - 2 C + D$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$0 = - 2 C + D$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $C$ in $0 = - 2 C + D$ with $0$ .

$0 = - 2 \cdot 0 + D$

$C = 0$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $- 2 \cdot 0 + D$ .

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Step 1.3.2.2.1.1

Multiply $- 2$ by $0$ .

$0 = 0 + D$

$C = 0$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

Step 1.3.2.2.1.2

Add $0$ and $D$ .

$0 = D$

$C = 0$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

$0 = D$

$C = 0$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

$0 = D$

$C = 0$

$3 = A + 6 C - 2 D$

$- 3 = B + 6 D$

Step 1.3.2.3

Replace all occurrences of $C$ in $3 = A + 6 C - 2 D$ with $0$ .

$3 = A + 6 (0) - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify $A + 6 (0) - 2 D$ .

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Step 1.3.2.4.1.1

Multiply $6$ by $0$ .

$3 = A + 0 - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.2.4.1.2

Add $A$ and $0$ .

$3 = A - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

$3 = A - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

$3 = A - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

$3 = A - 2 D$

$0 = D$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.3

Rewrite the equation as $D = 0$ .

$D = 0$

$3 = A - 2 D$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.4

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

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Step 1.3.4.1

Replace all occurrences of $D$ in $3 = A - 2 D$ with $0$ .

$3 = A - 2 \cdot 0$

$D = 0$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $A - 2 \cdot 0$ .

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Step 1.3.4.2.1.1

Multiply $- 2$ by $0$ .

$3 = A + 0$

$D = 0$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.4.2.1.2

Add $A$ and $0$ .

$3 = A$

$D = 0$

$C = 0$

$- 3 = B + 6 D$

$3 = A$

$D = 0$

$C = 0$

$- 3 = B + 6 D$

$3 = A$

$D = 0$

$C = 0$

$- 3 = B + 6 D$

Step 1.3.4.3

Replace all occurrences of $D$ in $- 3 = B + 6 D$ with $0$ .

$- 3 = B + 6 (0)$

$3 = A$

$D = 0$

$C = 0$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $B + 6 (0)$ .

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Step 1.3.4.4.1.1

Multiply $6$ by $0$ .

$- 3 = B + 0$

$3 = A$

$D = 0$

$C = 0$

Step 1.3.4.4.1.2

Add $B$ and $0$ .

$- 3 = B$

$3 = A$

$D = 0$

$C = 0$

$- 3 = B$

$3 = A$

$D = 0$

$C = 0$

$- 3 = B$

$3 = A$

$D = 0$

$C = 0$

$- 3 = B$

$3 = A$

$D = 0$

$C = 0$

Step 1.3.5

Rewrite the equation as $B = - 3$ .

$B = - 3$

$3 = A$

$D = 0$

$C = 0$

Step 1.3.6

Rewrite the equation as $A = 3$ .

$A = 3$

$B = - 3$

$D = 0$

$C = 0$

Step 1.3.7

List all of the solutions.

$A = 3, B = - 3, D = 0, C = 0$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A u + B}{{(u^{2} - 2 u + 6)}^{2}} + \frac{C u + D}{u^{2} - 2 u + 6}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{3 u - 3}{{(u^{2} - 2 u + 6)}^{2}} + \frac{0 u + 0}{u^{2} - 2 u + 6}$

Step 1.5

Simplify.

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Step 1.5.1

Factor $3$ out of $3 \cdot u - 3$ .

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Step 1.5.1.1

Factor $3$ out of $3 \cdot u$ .

$\frac{3 (u) - 3}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(0) \cdot u + 0}{u^{2} - 2 u + 6}$

Step 1.5.1.2

Factor $3$ out of $- 3$ .

$\frac{3 (u) + 3 (- 1)}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(0) \cdot u + 0}{u^{2} - 2 u + 6}$

Step 1.5.1.3

Factor $3$ out of $3 (u) + 3 (- 1)$ .

$\frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}} + \frac{(0) \cdot u + 0}{u^{2} - 2 u + 6}$

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

Multiply $0$ by $u$ .

$\frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}} + \frac{0 + 0}{u^{2} - 2 u + 6}$

Step 1.5.2.2

Add $0$ and $0$ .

$\frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}} + \frac{0}{u^{2} - 2 u + 6}$

Step 1.5.3

Divide $0$ by $u^{2} - 2 u + 6$ .

$\frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}} + 0$

Step 1.5.4

Remove the zero from the expression.

$\int \frac{3 (u - 1)}{{(u^{2} - 2 u + 6)}^{2}} d u$

Step 2

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

$3 \int \frac{u - 1}{{(u^{2} - 2 u + 6)}^{2}} d u$

Step 3

Let $u_{1} = u^{2} - 2 u + 6$ . Then $d u_{1} = (2 u - 2) d u$ , so $\frac{1}{2} d u_{1} = (u - 1) d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = u^{2} - 2 u + 6$ . Find $\frac{d u_{1}}{d u}$ .

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Step 3.1.1

Differentiate $u^{2} - 2 u + 6$ .

$\frac{d}{d u} [u^{2} - 2 u + 6]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $u^{2} - 2 u + 6$ with respect to $u$ is $\frac{d}{d u} [u^{2}] + \frac{d}{d u} [- 2 u] + \frac{d}{d u} [6]$ .

$\frac{d}{d u} [u^{2}] + \frac{d}{d u} [- 2 u] + \frac{d}{d u} [6]$

Step 3.1.2.2

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

$2 u + \frac{d}{d u} [- 2 u] + \frac{d}{d u} [6]$

Step 3.1.3

Evaluate $\frac{d}{d u} [- 2 u]$ .

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Step 3.1.3.1

Since $- 2$ is constant with respect to $u$ , the derivative of $- 2 u$ with respect to $u$ is $- 2 \frac{d}{d u} [u]$ .

$2 u - 2 \frac{d}{d u} [u] + \frac{d}{d u} [6]$

Step 3.1.3.2

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

$2 u - 2 \cdot 1 + \frac{d}{d u} [6]$

Step 3.1.3.3

Multiply $- 2$ by $1$ .

$2 u - 2 + \frac{d}{d u} [6]$

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

Since $6$ is constant with respect to $u$ , the derivative of $6$ with respect to $u$ is $0$ .

$2 u - 2 + 0$

Step 3.1.4.2

Add $2 u - 2$ and $0$ .

$2 u - 2$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$3 \int \frac{1}{{u_{1}}^{2}} \cdot \frac{1}{2} d u_{1}$

Step 4

Simplify.

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Step 4.1

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

$3 \int \frac{1}{{u_{1}}^{2} \cdot 2} d u_{1}$

Step 4.2

Move $2$ to the left of ${u_{1}}^{2}$ .

$3 \int \frac{1}{2 {u_{1}}^{2}} d u_{1}$

Step 5

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

$3 (\frac{1}{2} \int \frac{1}{{u_{1}}^{2}} d u_{1})$

Step 6

Combine fractions.

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Step 6.1

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} \int \frac{1}{{u_{1}}^{2}} d u_{1}$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$\frac{3}{2} \int {({u_{1}}^{2})}^{- 1} d u_{1}$

Step 6.2.2

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

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Step 6.2.2.1

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

$\frac{3}{2} \int {u_{1}}^{2 \cdot - 1} d u_{1}$

Step 6.2.2.2

Multiply $2$ by $- 1$ .

$\frac{3}{2} \int {u_{1}}^{- 2} d u_{1}$

Step 7

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

$\frac{3}{2} (- {u_{1}}^{- 1} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{3}{2} (- {u_{1}}^{- 1} + C)$ as $\frac{3}{2} (- \frac{1}{u_{1}}) + C$ .

$\frac{3}{2} (- \frac{1}{u_{1}}) + C$

Step 8.2

Multiply $\frac{3}{2}$ by $\frac{1}{u_{1}}$ .

$- \frac{3}{2 u_{1}} + C$

Step 9

Replace all occurrences of $u_{1}$ with $u^{2} - 2 u + 6$ .

$- \frac{3}{2 (u^{2} - 2 u + 6)} + C$