K2KARAKORAM
English
Open the app

CBSE · Class 12 · Mathematics

Integration by Partial Fractions (Ch 7: 7.5)

2 concepts · 16 practice questions with answers · Free to read

Distinct linear factors

From daily life

A big parcel is easier to carry in two small bags. A fraction with a product of factors below is split into two simple fractions, and each is integrated on its own.

Simple definition

If the denominator has distinct linear factors, write the fraction as a sum with one constant over each factor. Multiply by the denominator and substitute the root of each factor to get the constants quickly. An improper fraction must be divided first.

Formula: (px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b)

Rule 1: Split and find the constants — Substitute x = a and x = b.

  • Easy: 1/((x − 1)(x − 2)) = 1/(x − 2) − 1/(x − 1), so the integral is ln|(x − 2)/(x − 1)| + C.
  • Medium: (3x + 5)/((x + 1)(x + 3)) = 1/(x + 1) + 2/(x + 3), so the integral is ln|x + 1| + 2 ln|x + 3| + C.
  • Exam-type: (x + 1)/((x − 2)(x − 3)) = −3/(x − 2) + 4/(x − 3), giving −3 ln|x − 2| + 4 ln|x − 3| + C.

Rule 2: Improper fraction — Divide first, then split the remainder.

  • Easy: (x² + 1)/(x + 1) = x − 1 + 2/(x + 1), so the integral is x²/2 − x + 2 ln|x + 1| + C.
  • Medium: x²/(x² − 1) = 1 + ½ [1/(x − 1) − 1/(x + 1)], so the integral is x + ½ ln|(x − 1)/(x + 1)| + C.
  • Exam-type: x³/(x² − 1) = x + ½ [1/(x − 1) + 1/(x + 1)], so the integral is x²/2 + ½ ln|x² − 1| + C.
Remember

The degree of the numerator must be less than the degree of the denominator before splitting.

Common mistake

Splitting an improper fraction without dividing, which makes the constants impossible to find.

Memory clue

Cover the factor, put in its root, read the constant.

Practice questions

Easy
  1. 1Find ∫ dx/((x + 1)(x + 2)).
  2. 2Find ∫ (2x + 1)/((x − 1)(x + 2)) dx.
  3. 3Find ∫ x/((x + 1)(x + 2)) dx.
Medium
  1. 4Find ∫ dx/(x(x + 3)).
  2. 5Find ∫ (x + 7)/((x − 1)(x + 3)) dx.
  3. 6Find ∫ (x² + 1)/(x + 1) dx.
Challenging
  1. 7Find ∫ x²/(x² − 1) dx.
  2. 8Find ∫ (3x + 1)/(x² − 1) dx.
Show answers (try them yourself first)
  1. 1ln|(x + 1)/(x + 2)| + C
  2. 2ln|x − 1| + ln|x + 2| + C = ln|(x − 1)(x + 2)| + C
  3. 3−ln|x + 1| + 2 ln|x + 2| + C
  4. 4(1/3) ln|x/(x + 3)| + C
  5. 52 ln|x − 1| − ln|x + 3| + C
  6. 6x²/2 − x + 2 ln|x + 1| + C
  7. 7x + ½ ln|(x − 1)/(x + 1)| + C
  8. 82 ln|x − 1| + ln|x + 1| + C

Repeated and quadratic factors

From daily life

Some jobs repeat. A repeated factor needs one fraction for each power, and a factor like x² + 1 that cannot be broken needs a linear numerator.

Simple definition

For a repeated factor (x − a)² use A/(x − a) + B/(x − a)². For an irreducible quadratic factor use (Cx + D)/(x² + k²). Compare coefficients to find all the constants.

Formula: 1/((x − a)(x² + b)) = A/(x − a) + (Bx + C)/(x² + b)

Rule 1: Repeated factor — One term for each power.

  • Easy: x/(x + 1)² = 1/(x + 1) − 1/(x + 1)², so the integral is ln|x + 1| + 1/(x + 1) + C.
  • Medium: (3x + 2)/(x − 1)² = 3/(x − 1) + 5/(x − 1)², so the integral is 3 ln|x − 1| − 5/(x − 1) + C.
  • Exam-type: 1/(x(x + 1)²) = 1/x − 1/(x + 1) − 1/(x + 1)², so the integral is ln|x| − ln|x + 1| + 1/(x + 1) + C.

Rule 2: Quadratic factor — The numerator over x² + k² is a linear expression.

  • Easy: 1/(x(x² + 1)) = 1/x − x/(x² + 1), so the integral is ln|x| − ½ ln(x² + 1) + C.
  • Medium: 1/((x − 1)(x² + 1)) = ½ · 1/(x − 1) − ½ · (x + 1)/(x² + 1), so the integral is ½ ln|x − 1| − ¼ ln(x² + 1) − ½ tan⁻¹x + C.
  • Exam-type: x/((x − 1)(x² + 1)) = ½ · 1/(x − 1) + ½ · (1 − x)/(x² + 1), so the integral is ½ ln|x − 1| + ½ tan⁻¹x − ¼ ln(x² + 1) + C.
Remember

A square factor needs two terms, and an unbreakable quadratic needs a numerator with x.

Common mistake

Writing only A/(x − a)² for a repeated factor and leaving out A/(x − a).

Memory clue

Repeat the power, and give the quadratic an x.

Practice questions

Easy
  1. 1Find ∫ x/(x + 1)² dx.
  2. 2Find ∫ (3x + 2)/(x − 1)² dx.
  3. 3Find ∫ dx/(x(x² + 1)).
Medium
  1. 4Find ∫ dx/((x − 1)(x² + 1)).
  2. 5Find ∫ (2x + 1)/(x + 2)² dx.
  3. 6Find ∫ x/((x − 1)(x² + 1)) dx.
Challenging
  1. 7Find ∫ dx/(x(x + 1)²).
  2. 8Write the form of the partial fractions of (x + 3)/((x − 1)²(x² + 4)). Do not find the constants.
Show answers (try them yourself first)
  1. 1ln|x + 1| + 1/(x + 1) + C
  2. 23 ln|x − 1| − 5/(x − 1) + C
  3. 3ln|x| − ½ ln(x² + 1) + C
  4. 4½ ln|x − 1| − ¼ ln(x² + 1) − ½ tan⁻¹x + C
  5. 52 ln|x + 2| + 3/(x + 2) + C
  6. 6½ ln|x − 1| + ½ tan⁻¹x − ¼ ln(x² + 1) + C
  7. 7ln|x| − ln|x + 1| + 1/(x + 1) + C
  8. 8A/(x − 1) + B/(x − 1)² + (Cx + D)/(x² + 4)