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Indefinite Integration Formulas

Integration by substitution, integration by parts (ILATE), partial fractions, and standard trigonometric integrals.

9 Formulas•765+ Questions in Bank

Core Formulas & Mathematical Identities

9
01

Integration by Parts (ILATE Rule)

∫uv dx=u∫v dx−∫(u′∫v dx)dx\int u v \, dx = u \int v \, dx - \int \left( u' \int v \, dx \right) dx

Integration of product of two functions with priority order: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

Examiner Tip: For ∫ ln x dx, treat as ∫ (ln x) · 1 dx (u = ln x, v = 1).
02

Special Exponential Cancellation Integral

∫ex[f(x)+f′(x)] dx=exf(x)+C,∫[f(x)+xf′(x)] dx=xf(x)+C\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C, \quad \int [f(x) + x f'(x)] \, dx = x f(x) + C

Classic JEE pattern where derivative cancels out directly.

Examiner Tip: Always try to express awkward integrals in the form e^x [f(x) + f'(x)].
03

Standard Algebraic Quadratic Integrals

∫dxx2−a2=12aln⁡∣x−ax+a∣+C,∫dxx2±a2=ln⁡∣x+x2±a2∣+C\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C, \quad \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C

Standard formulas for quadratic denominators.

Examiner Tip: ∫ dx / (a² + x²) = (1/a) tan⁻¹(x/a) + C.
04

Trigonometric Substitution Guidelines

a2−x2  ⟹  x=asin⁡θ;a2+x2  ⟹  x=atan⁡θ;x2−a2  ⟹  x=asec⁡θ\sqrt{a^2 - x^2} \implies x = a\sin\theta; \quad \sqrt{a^2 + x^2} \implies x = a\tan\theta; \quad \sqrt{x^2 - a^2} \implies x = a\sec\theta

Standard substitutions that eliminate radicals via trigonometric identities.

Examiner Tip: For √( (a - x)/(a + x) ), substitute x = a cos 2θ.
05

Three Fundamental Radical Integrals

∫a2−x2dx=x2a2−x2+a22sin⁡−1xa+C,∫x2±a2dx=x2x2±a2±a22ln⁡∣x+x2±a2∣+C\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C, \quad \int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln|x + \sqrt{x^2 \pm a^2}| + C

Indispensable formulas for area calculations and conic integrals in JEE Advanced.

Examiner Tip: Memory trick: All three start with (x/2)·[original integrand], followed by (a²/2)·[integral of 1/[original integrand]].
06

Symmetric Biquadratic Rational Integrals

∫x2±1x4+kx2+1dx=∫1±1/x2(x∓1/x)2+(k±2)dx\int \frac{x^2 \pm 1}{x^4 + kx^2 + 1} dx = \int \frac{1 \pm 1/x^2}{(x \mp 1/x)^2 + (k \pm 2)} dx

Divide numerator and denominator by x² and substitute t = x ∓ 1/x.

Examiner Tip: If numerator is 1 or x², express as 1/2 · [(x² + 1) ± (x² - 1)].
07

Linear over Quadratic & Radical Integrals

∫px+qax2+bx+cdx=p2a∫2ax+bax2+bx+cdx+(q−pb2a)∫dxax2+bx+c\int \frac{px + q}{ax^2 + bx + c} dx = \frac{p}{2a} \int \frac{2ax + b}{ax^2 + bx + c} dx + \left(q - \frac{pb}{2a}\right) \int \frac{dx}{ax^2 + bx + c}

Split numerator into derivative of quadratic denominator plus a constant.

Examiner Tip: The first part integrates directly to (p/2a) ln|ax² + bx + c|, while the second is resolved by completing the square.
08

Walli's Formula (Definite Integral Reduction)

∫0π/2sin⁡nx dx=∫0π/2cos⁡nx dx={(n−1)!!n!!⋅π2n even(n−1)!!n!!n odd\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx = \begin{cases} \dfrac{(n-1)!!}{n!!} \cdot \dfrac{\pi}{2} & n \text{ even} \\[6pt] \dfrac{(n-1)!!}{n!!} & n \text{ odd} \end{cases}

Walli's (Wallis) reduction for ∫₀^{π/2} sinⁿx dx. The double-factorial n!! = n·(n-2)·(n-4)···. E.g., ∫₀^{π/2} sin⁴x dx = (3·1)/(4·2) · π/2 = 3π/16.

Examiner Tip: Quick check: ∫₀^{π/2} sin²x dx = π/4, ∫₀^{π/2} sin³x dx = 2/3, ∫₀^{π/2} sin⁵x dx = 8/15. Walli's eliminates repeated IBP in JEE Advanced.
09

Reduction Formulas for sin^n x and cos^n x

∫sin⁡nx dx=−sin⁡n−1xcos⁡xn+n−1n∫sin⁡n−2x dx,∫cos⁡nx dx=cos⁡n−1xsin⁡xn+n−1n∫cos⁡n−2x dx\int \sin^n x \, dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2}x \, dx, \quad \int \cos^n x \, dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2}x \, dx

Recursive IBP-based formulas to reduce power n by 2 each step until reaching ∫sin x dx or ∫dx. Terminates at base cases n = 0 (→ x) or n = 1 (→ -cos x / sin x).

Examiner Tip: Also: ∫ sec³x dx = (1/2)[sec x tan x + ln|sec x + tan x|] + C — needed in arc-length and surface-area integrals.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Indefinite Integration

  • •Forgetting the arbitrary constant of integration + C.
  • •Integration by parts requires correct priority order ILATE (Inverse, Log, Algebraic, Trig, Exponential).
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