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CAPF AC Integration

Study Material — 1 PYQs (2019–2019) · Concept Notes · Shortcuts

CAPF AC Integration is a frequently tested subtopic — 1 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.

1 PYQs
2019–2019
35 Practice
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10 Key Points
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Previous Year Questions

CAPF AC Integration — Past Exam Questions

1 questions from actual CAPF AC papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

Evaluate the definite integral: ∫₀^(π/4) x·sec²(x) dx

Concept Notes

Integration— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Integration is the reverse process of differentiation. Think of it as finding the original function when you know its derivative. If differentiation tells you the rate of change, integration tells you the total accumulation. Core Concept: Integration finds the area under a curve or the antiderivative of a function. The symbol ∫ represents integration. When we write ∫f(x)dx, we mean 'find the function whose derivative is f(x)'.

2.Key Rules

Core rules you must know cold
Power Rule

∫x^n dx = x^(n+1)/(n+1) + C (where n ≠ -1) 2

Constant Rule

∫k dx = kx + C 3

Sum Rule

∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx 4

Constant Multiple Rule

∫k·f(x)dx = k∫f(x)dx

3.Formula Block

Memorise — at least one formula appears in every paper
∫1 dx = x + C
∫x dx = x²/2 + C
∫x² dx = x³/3 + C
∫1/x dx = ln|x| + C
∫e^x dx = e^x + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec²x dx = tan x + C

Block:

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks 2-3 integration questions worth 7-10 marks. Common question types include basic integration using standard formulas, integration by substitution, and definite integration for area calculation. Questions often combine algebraic manipulation with integration rules.

5.Shortcuts

Use these to save 30–60 seconds per question

#1 - LIATE Rule for Integration by Parts: When using ∫u dv = uv - ∫v du, choose 'u' in this priority order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential functions. Shortcut Trick #2 - Quick Power Rule: For ∫x^n dx, add 1 to power, divide by new power. So ∫x⁵ dx becomes x⁶/6 + C instantly.

6.Worked Example

Solve this step-by-step before moving on
1
Step 1

Apply sum rule to separate terms = ∫3x²dx + ∫2x dx + ∫(-5)dx

2
Step 2

Take out constants = 3∫x²dx + 2∫x dx - 5∫1 dx

3
Step 3

Apply power rule to each term = 3(x³/3) + 2(x²/2) - 5x + C

4
Step 4

Simplify = x³ + x² - 5x + C Worked Example 2: Evaluate definite integral ∫₀² (x² + 1)dx

1
Step 1

Find indefinite integral first ∫(x² + 1)dx = x³/3 + x + C

2
Step 2

Apply limits using fundamental theorem = [x³/3 + x]₀²

3
Step 3

Substitute upper limit minus lower limit = (2³/3 + 2) - (0³/3 + 0) = 8/3 + 2 - 0 = 8/3 + 6/3 = 14/3 Shortcut Trick #3 - Pattern Recognition: If you see (ax + b)^n in the integrand, use substitution u = ax + b. The answer pattern becomes (ax + b)^(n+1)/[a(n+1)] + C. Most Common Trap: Students forget the constant of integration '+C' in indefinite integrals. This costs valuable marks. In definite integrals, the constant cancels out, but in indefinite integrals, it's mandatory. Another major mistake is incorrect application of limits in definite integration - always substitute upper limit first, then subtract the lower limit value. Another frequent error occurs with the power rule when n = -1. Remember: ∫x^(-1) dx = ∫(1/x) dx = ln|x| + C, NOT x⁰/0. The power rule breaks down here, requiring the logarithmic form instead.

Key Points to Remember

  • Integration is the reverse of differentiation - finding original function from its derivative
  • Power rule formula: ∫x^n dx = x^(n+1)/(n+1) + C where n ≠ -1
  • Always add constant '+C' in indefinite integration
  • Sum rule: integral of sum equals sum of integrals
  • For definite integrals: substitute upper limit minus lower limit
  • Quick power rule: add 1 to power, divide by new power
  • ∫1/x dx = ln|x| + C (special case when power = -1)
  • LIATE rule for integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential
  • Substitution pattern: ∫(ax+b)^n dx = (ax+b)^(n+1)/[a(n+1)] + C
  • Standard integrals: ∫sin x dx = -cos x + C, ∫cos x dx = sin x + C

Exam-Specific Tips

  • ∫e^x dx = e^x + C (exponential function integrates to itself)
  • ∫sec²x dx = tan x + C (standard trigonometric integral)
  • ∫cosec²x dx = -cot x + C (negative cotangent result)
  • ∫1/√(1-x²) dx = sin⁻¹x + C (inverse sine integration)
  • ∫1/(1+x²) dx = tan⁻¹x + C (inverse tangent integration)
  • ∫sec x tan x dx = sec x + C (product rule reverse)
  • ∫cosec x cot x dx = -cosec x + C (negative cosecant result)
  • Fundamental theorem: d/dx[∫f(x)dx] = f(x) (differentiation and integration are inverse operations)
Practice MCQs

Integration — Practice Questions

35graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 35

All MCQs →
Practice 1easy

Find the area under the curve y = x² between x = 1 and x = 3.

Practice 2easy

Evaluate ∫ (2x + 3)/(x² + 3x + 2) dx

Practice 3easy

Find the area enclosed between the curves y = x² and y = 2x in the first quadrant.

Practice 4easy

Evaluate the definite integral: ∫₀^(π/2) sin(x) cos(x) dx

Practice 5easy

Find ∫ x e^(2x) dx using integration by parts.

Practice 6easy

Find ∫ (2x + 1)/(x² + x + 1) dx.

Practice 7easy

Evaluate ∫₀^(π/4) sec²(x) dx.

Practice 8easy

Find ∫₀^(π/2) cos x dx

Practice 9easy

Evaluate ∫ x·e^x dx using integration by parts.

Practice 10easy

Evaluate the indefinite integral: ∫(3x² + 2sin x − e^x) dx

Practice 11easy

Evaluate the indefinite integral: ∫(3x² + 2sin x - e^x) dx

Practice 12medium

Evaluate ∫₀^(π/4) tan²(x) dx

Practice 13medium

Evaluate ∫₁² (2x³ − 3x² + 5x − 2)/(x − 1) dx

Practice 14medium

Find ∫ x·e^(2x) dx

Practice 15medium

Evaluate ∫₁² (x² − 1)/x² dx

Practice 16medium

Find the area enclosed between the curves y = x² and y = 2x in the first quadrant.

Practice 17medium

Evaluate ∫ (sin(x))/(1 + cos(x)) dx

Practice 18medium

Evaluate ∫₁² (x² − 1)/(x³ − 3x + 2) dx.

Practice 19medium

Compute ∫₀^(π/4) tan(x) dx.

Practice 20medium

Find the area enclosed between the curves y = x² and y = 2x − x² from x = 0 to x = 1.

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60-Second Revision — Integration

  • Formula: ∫x^n dx = x^(n+1)/(n+1) + C, remember the +C always
  • Trap: Never forget constant of integration in indefinite integrals
  • Remember: ∫1/x dx = ln|x| + C, not power rule
  • Quick method: For (ax+b)^n, answer is (ax+b)^(n+1)/[a(n+1)] + C
  • Definite integrals: [F(x)]ₐᵇ = F(b) - F(a), upper minus lower
  • Standard results: ∫sin x = -cos x, ∫cos x = sin x, ∫e^x = e^x
  • LIATE priority for integration by parts: L-I-A-T-E order
←Applications of DerivativesDefinite Integrals & Applications→
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