The Chapter in One Sweep

Integration is differentiation run backwards. If ddxF(x)=f(x)\frac{d}{dx}F(x) = f(x), then ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C — the indefinite integral, a whole family of functions differing by the constant CC. The definite integral ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a) is a single number.

The standard integrals table (know it cold)

  1. ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C (n≠−1n \neq -1); ∫dxx=log⁡∣x∣+C\int\frac{dx}{x} = \log|x| + C
  2. ∫cos⁡x dx=sin⁡x+C\int\cos x\,dx = \sin x + C; ∫sin⁡x dx=−cos⁡x+C\int\sin x\,dx = -\cos x + C
  3. ∫sec⁡2x dx=tan⁡x+C\int\sec^2x\,dx = \tan x + C; ∫cosec2 x dx=−cot⁡x+C\int\mathrm{cosec}^2\,x\,dx = -\cot x + C
  4. ∫sec⁡xtan⁡x dx=sec⁡x+C\int\sec x\tan x\,dx = \sec x + C; ∫cosec xcot⁡x dx=−cosec x+C\int\mathrm{cosec}\,x\cot x\,dx = -\mathrm{cosec}\,x + C
  5. ∫dx1−x2=sin⁡−1x+C\int\frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}x + C; ∫dx1+x2=tan⁡−1x+C\int\frac{dx}{1 + x^2} = \tan^{-1}x + C
  6. ∫exdx=ex+C\int e^x dx = e^x + C; ∫axdx=axlog⁡a+C\int a^x dx = \frac{a^x}{\log a} + C

The four famous results (proved by substitution)

Card of the four famous integrals of tan cot sec cosec

The technique decision tree

  1. Composite function with its inner derivative visible → substitution t=g(x)t = g(x).
  2. Rational function → proper? decompose by partial fractions (five templates). Improper? divide first.
  3. Product of unlike functions → by parts, first function by ILATE; watch for the ex[f+f′]e^x[f + f'] shortcut.
  4. Quadratic (or its root) in the denominator → complete the square, match one of the six standard forms.
  5. quadratic\sqrt{\text{quadratic}} being integrated → the three root formulas.

Card of six standard integral forms with quadratic denominators

Templates and Definite-Integral Toolkit

Partial fraction templates (Table 7.2)

Table of five partial fraction templates

By parts and the root integrals

∫f(x)g(x) dx=f(x)∫g(x) dx−∫[f′(x)∫g(x) dx]dx\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int\left[f'(x)\int g(x)\,dx\right]dx

Special patterns: ∫ex[f(x)+f′(x)]dx=exf(x)+C\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + C, and the three root formulas:

Card of the three standard square root integrals

Definite integrals

  1. Second FTC: find any anti-derivative FF (drop CC), compute F(b)−F(a)F(b) - F(a). The integrand must be defined and continuous on all of [a,b][a, b].
  2. Substitution: move the limits with the variable — never evaluate a tt-antiderivative at xx-limits.
  3. Properties:

Card listing properties P0 to P7 of definite integrals

The three most-used values: ∫0π/2log⁡sin⁡x dx=−π2log⁡2\int_0^{\pi/2}\log\sin x\,dx = -\frac{\pi}{2}\log 2; ∫0π/4log⁡(1+tan⁡x)dx=π8log⁡2\int_0^{\pi/4}\log(1 + \tan x)dx = \frac{\pi}{8}\log 2; ∫0π/2sin⁡nxsin⁡nx+cos⁡nxdx=π4\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx + \cos^nx}dx = \frac{\pi}{4} for every nn.

The Mistake Checklist — Read Before Every Exam

  1. Dropped +C+C on an indefinite integral — an instant half-mark gone.
  2. Chain-rule amnesia: ∫cos⁡2x dx=sin⁡2x2\int\cos 2x\,dx = \frac{\sin 2x}{2}, not sin⁡2x\sin 2x. Divide by the inner derivative for every linear inner function.
  3. Power rule on 1x\frac1x: ∫dxx=log⁡∣x∣\int\frac{dx}{x} = \log|x| — the n=−1n = -1 case is the exception to xn+1n+1\frac{x^{n+1}}{n+1}.
  4. Sign slips with t=cos⁡xt = \cos x: dt=−sin⁡x dxdt = -\sin x\,dx — the minus must appear.
  5. Forgetting to move limits after substitution in a definite integral.
  6. sin⁡2x\sin^2 x treated as odd: squares of odd functions are even. Check the whole integrand's parity, and confirm the interval is symmetric before using P7.
  7. Improper fractions decomposed directly: divide first when deg⁡P≥deg⁡Q\deg P \geq \deg Q.
  8. Modulus integrated without splitting: find the sign changes, split by P2, then integrate.
  9. Wrong first function in by parts: ILATE — inverse trig and logs before algebra; algebra before trig and exponentials.
  10. FTC across a discontinuity: check the integrand exists on the whole interval (NCERT's ∫−23x(x2−1)1/2dx\int_{-2}^{3}x\left(x^2 - 1\right)^{1/2}dx warning).
  11. Mixing the six standard forms: the sign pattern of the quadratic (x2−a2x^2 - a^2, a2−x2a^2 - x^2, x2+a2x^2 + a^2) picks the formula — memorise them as three pairs (with root, without root).
  12. King's rule missed: if xx multiplies a symmetric trig block on [0,π][0, \pi] or [0,π2][0, \frac{\pi}{2}], reflect and add before attempting anything else.

Key Point: Revision strategy: one pass through the formula cards above, one pass through this checklist, then attempt the Section 12 drill under time. Wrong answers point to the section to re-read.

[JEE Tip] The night before the exam, re-derive (don't just re-read) the four famous integrals, one root formula, and one King's-rule evaluation — active recall of one representative per family locks in the whole family.