Integrals

Why This Chapter Matters

Integration is the biggest chapter in JEE calculus — 10-15 marks. Standard integrals, integration by parts, definite integrals, and area under curves are all tested. Practice is the only way to master this.

Core Concepts

1. Standard Integrals (Must Memorize)

integral x^n dx = x^(n+1)/(n+1) + C (n != -1)

integral 1/x dx = ln|x| + C

integral e^x dx = e^x + C

integral a^x dx = a^x/ln(a) + C

integral sin x dx = -cos x + C

integral cos x dx = sin x + C

integral tan x dx = ln|sec x| + C = -ln|cos x| + C

integral cot x dx = ln|sin x| + C

integral sec x dx = ln|sec x + tan x| + C

integral cosec x dx = ln|cosec x - cot x| + C

integral sec^2 x dx = tan x + C

integral cosec^2 x dx = -cot x + C

integral sec x tan x dx = sec x + C

integral 1/(1+x^2) dx = tan^-1 x + C

integral 1/sqrt(1-x^2) dx = sin^-1 x + C

2. Methods of Integration

Substitution: Replace u = g(x), du = g'(x)dx

integral f(g(x)) g'(x) dx = integral f(u) du

Integration by Parts (ILATE rule):

integral u v dx = u x integral(v dx) - integral [u' x integral(v dx)] dx

ILATE priority: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential

Partial Fractions:

For rational functions, decompose into simpler fractions.

1/(x(x+1)) = 1/x - 1/(x+1)

3. Definite Integrals

integral from a to b of f(x) dx = F(b) - F(a) where F'(x) = f(x)

Properties:

•integral a to b of f(x)dx = -integral b to a of f(x)dx

•integral a to b of f(x)dx = integral a to c of f(x)dx + integral c to b of f(x)dx

•integral 0 to a of f(x)dx = integral 0 to a of f(a-x)dx [KEY property]

•integral -a to a of f(x)dx = 2 x integral 0 to a of f(x)dx if f is even

•integral -a to a of f(x)dx = 0 if f is odd

4. Area Under Curves

Area between y=f(x) and x-axis from a to b:

A = integral from a to b of |f(x)| dx

Area between two curves y=f(x) and y=g(x):

A = integral from a to b of |f(x) - g(x)| dx

PYQs

2024: integral of [sin x/(sin x + cos x)] dx from 0 to pi/2

Use property: I = integral of [cos x/(sin x + cos x)] dx (substitute pi/2-x)

2I = integral of 1 dx from 0 to pi/2 = pi/2. So I = pi/4.

2023: integral of x e^x dx

By parts (u=x, v=e^x): x e^x - integral e^x dx = x e^x - e^x + C = e^x(x-1) + C

2022: integral of 1/(x^2 + 4x + 5) dx

Complete the square: (x+2)^2 + 1. integral 1/((x+2)^2+1) dx = tan^-1(x+2) + C

MCQ Practice

Q1. integral of e^(3x) dx:

(A) e^(3x) (B) 3e^(3x) (C) e^(3x)/3 (D) 3e^(3x)+C

Answer: C

Q2. integral from 0 to pi of sin x dx:

(A) 0 (B) 1 (C) 2 (D) -2

Answer: C [[-cos x] from 0 to pi = -cos(pi)+cos(0) = 1+1 = 2]

Q3 (Hard). Area enclosed by y=x^2 and y=sqrt(x):

Intersection: x^2 = sqrt(x) => x^4 = x => x=0 or x=1.

Area = integral 0 to 1 of (sqrt(x) - x^2) dx = [2x^(3/2)/3 - x^3/3] from 0 to 1 = 2/3 - 1/3 = 1/3

Revision Notes

STANDARD: memorize all basic integrals listed above

SUBSTITUTION: replace u = inner function, find du
BY PARTS (ILATE): integral uv' = uv - integral u'v
PARTIAL FRACTIONS: decompose rational function

DEFINITE INTEGRAL PROPERTIES:
integral 0->a of f(x) = integral 0->a of f(a-x)  [USE THIS for symmetric integrals]
Even function: integral -a->a = 2 x integral 0->a
Odd function: integral -a->a = 0

AREA:
Under curve y=f(x): integral |f(x)| dx
Between curves: integral |f(x)-g(x)| dx  [from intersection to intersection]