Indefinite Integration

Indefinite Integral

• If F′(x) = f(x), then ∫ f(x) dx = F(x) + C
• C is the constant of integration

Basic Theorems of Integration

• ∫ k f(x) dx = k ∫ f(x) dx
• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
• d/dx [ ∫ f(x) dx ] = f(x)
• ∫ d/dx [f(x)] dx = f(x) + C

Standard Integrals

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
• ∫ 1/x dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ aˣ dx = aˣ / ln a + C
• ∫ sin x dx = −cos x + C
• ∫ cos x dx = sin x + C
• ∫ tan x dx = ln|sec x| + C
• ∫ cot x dx = ln|sin x| + C
• ∫ sec x dx = ln|sec x + tan x| + C
• ∫ cosec x dx = ln|cosec x − cot x| + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = −cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = −cosec x + C

Important Algebraic Forms

• ∫ 1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + C
• ∫ 1/(x² − a²) dx = (1/2a) ln|(x−a)/(x+a)| + C
• ∫ 1/√(a² − x²) dx = sin⁻¹(x/a) + C
• ∫ 1/√(x² + a²) dx = ln|x + √(x² + a²)| + C
• ∫ 1/√(x² − a²) dx = ln|x + √(x² − a²)| + C

Exponential–Trigonometric Integrals

• ∫ e^ax sin bx dx = e^ax(a sin bx − b cos bx)/(a² + b²) + C
• ∫ e^ax cos bx dx = e^ax(a cos bx + b sin bx)/(a² + b²) + C

Integration by Substitution

• Used when integrand contains a function and its derivative
• Put f(x) = t so that f′(x) dx = dt
• ∫ f(f(x)) f′(x) dx = ∫ f(t) dt
• If integrand is of form f(ax + b), put ax + b = t

Useful Standard Substitutions

• √(a² − x²) → x = a sinθ
• √(a² + x²) → x = a tanθ or a sinhθ
• √(x² − a²) → x = a secθ or a coshθ
• √((a−x)/(a+x)) → x = a cosθ

Integration by Parts

• ∫ u v dx = u ∫ v dx − ∫ (du/dx)(∫ v dx) dx
• Choose u using ILATE rule:
• Inverse trigonometric → Logarithmic → Algebraic → Trigonometric → Exponential
• For only log or inverse trig functions, take the other function as 1

Special Results

• ∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C
• ∫ f′(x)/f(x) dx = ln|f(x)| + C
• ∫ f′(x)/√f(x) dx = 2√f(x) + C

Trigonometric Rational Integrals

• For integrals involving a sin x + b cos x, write:
• a = r cosθ, b = r sinθ
• Then denominator reduces to r sin(x + θ) or r cos(x + θ)

JEE Main Focus Tips

• Always try substitution before expansion
• Memorise standard results — direct questions are common
• Integration by parts is frequently tested with log and inverse trig
• Watch for expressions of the form f(x)+f′(x)
• Final answer must include + C