ExamHabit
Practise free

JEE Main 2016

JEE Main 2016 Mathematics question on Indefinite Integration

The integral ∫2x12+5x9(x5+x3+1)3 dx\int \dfrac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3}\, dx is equal to
  1. A.
    −x5(x5+x3+1)2+C\dfrac{-x^5}{(x^5 + x^3 + 1)^2} + C
  2. B.
    x102(x5+x3+1)2+C\dfrac{x^{10}}{2(x^5 + x^3 + 1)^2} + C
  3. C.
    x52(x5+x3+1)2+C\dfrac{x^5}{2(x^5 + x^3 + 1)^2} + C
  4. D.
    −x102(x5+x3+1)2+C\dfrac{-x^{10}}{2(x^5 + x^3 + 1)^2} + C
Show answer and solution

Correct option: B

Solution

Divide numerator and denominator by x¹⁵:

Numerator: (2x¹² + 5x⁹)/x¹⁵ = 2/x³ + 5/x⁶

Denominator: (x⁵ + x³ + 1)³/x¹⁵ = (1 + 1/x² + 1/x⁵)³

So the integral becomes ∫ (2/x³ + 5/x⁶) / (1 + x⁻² + x⁻⁵)³ dx.

Let t = 1 + x⁻² + x⁻⁵. Then dt/dx = -2x⁻³ - 5x⁻⁶, so dt = -(2/x³ + 5/x⁶)dx.

The integral becomes ∫ -dt/t³ = 1/(2t²) + C = 1/[2(1 + x⁻² + x⁻⁵)²] + C.

Multiply top and bottom by x¹⁰: this equals x¹⁰/[2(x⁵ + x³ + 1)²] + C.

[JEE Main 2016]

Practise Indefinite Integration the way it is asked

A timed set of real JEE Main Indefinite Integration questions, graded as the exam grades. No sign-up.

Start a Indefinite Integration test

More: all JEE Main Indefinite Integration questions · JEE Main 2016 · Mathematics chapter list