Integrals

part 1

Basic Integration

• Integration as inverse of differentiation
• Standard integrals and basic rules
• Integration by substitution
• Integration of trigonometric functions

part 2

Advanced Techniques

• Integration by partial fractions
• Integration by parts
• Integration of special functions
• Definite integral — definition using limits

part 3

Definite Integrals

• Properties of definite integrals
• Fundamental theorem of calculus
• Evaluation of definite integrals
• Application — area under a curve

∫xⁿ dx

$\frac{x^{n+1}}{n+1} + C$ (n ≠ −1)

★ board favourite

∫eˣ dx

$e^x + C$

★ board favourite

∫(1/x) dx

$\ln|x| + C$

★ board favourite

∫sin x dx

$-\cos x + C$

★ board favourite

∫cos x dx

$\sin x + C$

★ board favourite

∫sec²x dx

$\tan x + C$

★ board favourite

∫cosec²x dx

$-\cot x + C$

★ board favourite

∫dx/(x²+a²)

$\frac{1}{a}\tan^{-1}\frac{x}{a} + C$

★ board favourite

∫dx/√(a²−x²)

$\sin^{-1}\frac{x}{a} + C$

★ board favourite

Integration by parts (ILATE)

$\int u\cdot v\,dx = u\cdot\int v\,dx - \int\left(u'\cdot\int v\,dx\right)dx$

★ board favourite

Evaluate ∫₁³ (2x² + 5x) dx as the limit of a sum.

Evaluate ∫ (x)/((x−1)(x−2)(x−3)) dx using partial fractions.

Evaluate ∫ x² e^x dx and ∫ x sin x dx using integration by parts.

Evaluate ∫₀^π (x·tan x)/(sec x + tan x) dx.

Evaluate ∫₀^(π/2) (sin x)/(sin x + cos x) dx using properties of definite integrals.

Evaluate ∫ e^x (1 + sin x)/(1 + cos x) dx.

Evaluate: (a) ∫ sin³x cos²x dx, (b) ∫ 1/(x² + 2x + 5) dx

Evaluate ∫ √(x² + 4x + 8) dx.

Evaluate ∫₀^(π/2) (2 log sin x − log sin 2x) dx.

Find ∫ (1 − sin x)/(cos²x) dx.

Evaluate ∫ 1/(3 + 2 sin x + cos x) dx.

How many marks is Integrals worth in the Class 12 board exam?

Around 8 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Integration by parts (ILATE) and partial fractions — very high yield.

Is Integrals easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 6 hours to cover it well.

What are the important topics in Integrals?

The chapter covers 14 NCERT topics in 3 areas: Basic Integration; Advanced Techniques; Definite Integrals.

What questions come from Integrals in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Definite Integral as Limit of Sum, Partial Fractions, Integration by Parts, Properties of Definite Integrals — 1- to 5-mark questions.