CBSE Class 12 Mathematics: Calculus — Integration — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: Evaluate: ∫sin²x dx.
  Ans: Using identity sin²x = (1 - cos2x)/2: ∫sin²x dx = ∫(1 - cos2x)/2 dx = x/2 - sin2x/4 + C = x/2 - sin2x/4 + C.
• Q: Evaluate: ∫₀^π sin x dx.
  Ans: ∫₀^π sin x dx = [-cos x]₀^π = -cos π - (-cos 0) = -(-1) + 1 = 1 + 1 = 2.
• Q: Find the area enclosed between y = x² and y = x.
  Ans: Intersection: x² = x → x² - x = 0 → x(x-1) = 0 → x = 0, 1. For 0 ≤ x ≤ 1: x > x² (line above parabola). Area = ∫₀¹(x - x²)dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6 sq. units.

Long Answer / Application Questions (4-6 Marks)

• Q: Evaluate by partial fractions: ∫dx/[x(x+1)].
  Ans: 1/[x(x+1)] = A/x + B/(x+1). 1 = A(x+1) + Bx. Put x=0: A=1. Put x=-1: B=-1. ∫[1/x - 1/(x+1)]dx = ln|x| - ln|x+1| + C = ln|x/(x+1)| + C.

Exam Tips for This Chapter

• Revise all definitions and laws from Calculus — Integration — commonly asked as 1-2 mark questions
• Practice diagrams related to Calculus — Integration — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Calculus — Integration carries significant marks in the board exam