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What is the Integration by Partial Fractions for Repeated Linear Factors?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Integration by Partial Fractions for Repeated Linear Factors is a special technique to integrate complicated fractions. When the bottom part (denominator) of a fraction has a factor like (ax + b) that appears more than once (e.g., (ax + b)^2 or (ax + b)^3), we break it down into simpler fractions. This makes it much easier to find its integral.
Simple Example
Quick Example
Imagine you have a big thali with many dishes, and you want to know the total 'taste score' of one specific repeated dish, like extra paneer butter masala. Instead of trying to rate the whole thali at once, you break down the thali into individual dishes and rate each paneer portion separately. Similarly, this method breaks a complex fraction into simpler parts, like (A/(ax+b)) + (B/(ax+b)^2), making the 'total score' (integral) easier to calculate.
Worked Example
Step-by-Step
Let's integrate 1/((x+1)^2 * x).
Step 1: Set up the partial fractions. Since (x+1) is a repeated linear factor, we write it as A/x + B/(x+1) + C/(x+1)^2.
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Step 2: Find a common denominator and equate numerators. Multiply both sides by x(x+1)^2: 1 = A(x+1)^2 + Bx(x+1) + Cx.
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Step 3: Solve for A, B, and C by choosing convenient values of x.
If x = 0: 1 = A(0+1)^2 + B(0)(0+1) + C(0) => 1 = A. So, A = 1.
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Step 4: If x = -1: 1 = A(-1+1)^2 + B(-1)(-1+1) + C(-1) => 1 = 0 + 0 - C => C = -1.
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Step 5: To find B, pick another value, say x = 1: 1 = A(1+1)^2 + B(1)(1+1) + C(1) => 1 = A(4) + B(2) + C.
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Step 6: Substitute A=1 and C=-1 into the equation from Step 5: 1 = 1(4) + B(2) + (-1) => 1 = 4 + 2B - 1 => 1 = 3 + 2B => -2 = 2B => B = -1.
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Step 7: Now, substitute A, B, C back into the partial fractions: Integral of (1/x - 1/(x+1) - 1/(x+1)^2) dx.
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Step 8: Integrate each term: ln|x| - ln|x+1| - (-1/(x+1)) + C = ln|x| - ln|x+1| + 1/(x+1) + C.
Answer: ln|x| - ln|x+1| + 1/(x+1) + C
Why It Matters
This method is super important in fields like Engineering and Physics. For example, when designing circuits or predicting how a rocket's speed changes, you often encounter complex fractions. Engineers use this technique to break down those problems, making calculations manageable and helping them build amazing machines or understand how the universe works.
Common Mistakes
MISTAKE: Not setting up the partial fractions correctly for repeated factors, e.g., writing A/(ax+b)^2 instead of A/(ax+b) + B/(ax+b)^2. | CORRECTION: For a factor (ax+b)^n, you need 'n' terms: A1/(ax+b) + A2/(ax+b)^2 + ... + An/(ax+b)^n.
MISTAKE: Making calculation errors when solving for the constants A, B, C. This often happens when substituting values of x. | CORRECTION: Double-check your arithmetic after each substitution. It's helpful to choose x values that make terms zero first, then use other values.
MISTAKE: Forgetting to integrate the '1/(ax+b)^n' terms correctly. Students might use ln for all, even when n is not 1. | CORRECTION: Remember that integral of 1/(ax+b) is (1/a)ln|ax+b|, but integral of 1/(ax+b)^2 is (-1/a)(1/(ax+b)), and so on, using the power rule for integration.
Practice Questions
Try It Yourself
QUESTION: Set up the partial fraction decomposition for (2x+3)/(x^2 * (x-1)). | ANSWER: A/x + B/x^2 + C/(x-1)
QUESTION: If the partial fraction decomposition for (3x+1)/(x(x+1)^2) is A/x + B/(x+1) + C/(x+1)^2, find the value of A. | ANSWER: A = 1
QUESTION: Integrate (2x+1)/(x^2 * (x+1)) dx. | ANSWER: ln|x| - 1/x - ln|x+1| + C
MCQ
Quick Quiz
Which of these is the correct partial fraction setup for (x+5)/(x-2)^3?
A/(x-2)
A/(x-2) + B/(x-2)^2
A/(x-2) + B/(x-2)^2 + C/(x-2)^3
A/(x-2)^3
The Correct Answer Is:
C
For a repeated linear factor like (x-2)^3, you must include a term for each power from 1 up to 3. So, A/(x-2), B/(x-2)^2, and C/(x-2)^3 are all needed.
Real World Connection
In the Real World
Imagine you're an engineer at ISRO designing a new rocket. The equations describing the rocket's acceleration or fuel consumption might involve complex fractions with repeated factors. Using this integration technique helps you accurately calculate how much fuel is needed or how fast the rocket will go, ensuring a successful launch like Chandrayaan!
Key Vocabulary
Key Terms
INTEGRATION: Finding the antiderivative of a function, like finding the total distance from speed over time. | PARTIAL FRACTIONS: Breaking down a complex fraction into a sum of simpler fractions. | LINEAR FACTOR: A term like (ax+b) where x has a power of 1. | REPEATED FACTOR: A linear factor that appears more than once in the denominator, like (x-1)^2. | DENOMINATOR: The bottom part of a fraction.
What's Next
What to Learn Next
Great job understanding repeated linear factors! Next, you should explore 'Integration by Partial Fractions for Irreducible Quadratic Factors'. This builds on what you've learned and tackles fractions where the bottom part has factors that can't be easily broken down, opening up even more types of problems.
