Definite Integration Short Tricks (JEE Main)

King's Rule (Reflection)

$\int_a^b f(x),dx = \int_a^b f(a+b-x),dx$. Adding gives fast answers when f(x)+f(a+b−x) simplifies.

Even-Odd on Symmetric Interval

• $\int_{-a}^{a} f(x)dx = 2\int_0^a f$ if $f$ even, $= 0$ if $f$ odd.

Period Trick

• If $f$ has period $T$: $\int_0^{nT} f = n\int_0^T f$.

Zero–π/2 Sin/Cos Symmetry

• $\int_0^{\pi/2} \ln(\sin x),dx = \int_0^{\pi/2} \ln(\cos x),dx = -\tfrac{\pi}{2}\ln 2$.
• $\int_0^{\pi/2} \sin^n x,dx = \int_0^{\pi/2} \cos^n x,dx$ (Wallis).
• Wallis: for even $n$: $\tfrac{(n-1)(n-3)\ldots1}{n(n-2)\ldots2}\cdot\tfrac{\pi}{2}$; odd n: $\tfrac{(n-1)(n-3)\ldots2}{n(n-2)\ldots1}$.

Greatest Integer / Fractional

• $\int_0^n [x],dx = n(n-1)/2$ for integer $n$.
• $\int_0^1 {x},dx = 1/2$.

Standard Results

• $\int_0^\infty e^{-ax}dx = 1/a$.
• $\int_0^\infty \frac{\sin x}{x}dx = \pi/2$.
• $\int_0^{2\pi} \sin^2 x,dx = \pi = \int_0^{2\pi}\cos^2 x,dx$.

Newton–Leibniz / Leibniz Rule

$\frac{d}{dt}\int_{u(t)}^{v(t)} f(x)dx = f(v(t))v'(t) - f(u(t))u'(t)$.

Common Traps

• Discontinuous integrand: split at discontinuity, e.g., $\int_{-1}^{1} |x|,dx = 1$.
• Modulus + trig: use $\sin x > 0$ on $(0,\pi)$, etc.

Practice MCQs

Q1. $\int_0^{\pi/2} \sin^6 x,dx$ equals A) 5π/32 B) 3π/16 C) 15π/48 D) 5π/16

Q2. $\int_{-\pi}^{\pi} x^3 \cos x,dx$ equals A) 0 B) π C) 2π D) −2π

Q3. $\int_0^{\pi/2} \frac{\sin x}{\sin x+\cos x}dx$ equals A) π/4 B) π/2 C) π/8 D) 1

Q4. $\int_0^{10\pi} |\sin x|,dx$ equals A) 10 B) 20 C) 10π D) 20π

Q5. $\int_0^1 [3x]dx$ equals A) 0 B) 1 C) 3/2 D) 1/3

Q6. $\frac{d}{dx}\int_0^{x^2} \sin t,dt$ equals A) sin x² B) 2x sin x² C) cos x² D) 2x cos x²

Q7. $\int_0^{\pi/2}\frac{dx}{1+\tan^n x}$ equals A) 0 B) π/8 C) π/4 D) π/2

Q8. $\int_0^{\pi} x \sin x,dx$ equals A) π B) 2π C) π/2 D) 0

Q9. $\int_1^4 [\sqrt x]dx$ equals A) 2 B) 3 C) 4 D) 5

Q10. $\int_{-2}^{2}\frac{x^2}{1+e^x}dx$ equals A) 0 B) 4/3 C) 8/3 D) 2

Answers: 1-A ((5·3·1)/(6·4·2) · π/2 = 15/48 · π/2 = 5π/32.), 2-A (Odd function on symmetric interval.), 3-A (Add to reflected integral; result = π/4.), 4-B (10 periods × 2 = 20.), 5-B (Split at 1/3 and 2/3: 0+1/3+2/3=1.), 6-B (Differentiate upper limit chain rule.), 7-C (Any n → π/4 by King's rule.), 8-A (IBP: [−xcos x]₀^π + ∫cos x dx = π.), 9-C (1 to 4: [√x]=1 on [1,4), =2 at 4 → 1·3=3? Actually [√x]=1 on [1,4), integral 3; add point value 0 → 3. Set as 3.), 10-B (f(x)+f(−x) = x²; ∫₀² x² dx = 8/3? split half → 4/3.)

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