Integrals Cheat Sheet for KCET (2nd PUC)
1. Standard Indefinite Integrals
Memorize these for quick recall (most KCET ques ons
test direct applica on):
∫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
∫sec²(x) dx = tan(x) + C
∫cosec²(x) dx = -cot(x) + C
∫sec(x)tan(x) dx = sec(x) + C
∫cosec(x)cot(x) dx = -cosec(x) + C
∫1/√(1-x²) dx = sin⁻¹(x) + C
∫1/(1+x²) dx = tan⁻¹(x) + C
∫1/|x|√(x²-1) dx = sec⁻¹(x) + C
KCET Tip: Ques ons o en give you the integrand
directly (e.g., ∫cos(x) dx). Spot the form and write the
answer in 10 seconds.
2. Subs tu on Method (Quick Tricks)
Many KCET integrals need a simple subs tu on to simplify. Look for these
pa erns:
Form: ∫f(g(x))g'(x) dx
o Subs tute u = g(x), so du = g'(x) dx.
o Example: ∫2x e^(x²) dx
Let u = x², du = 2x dx
Integral becomes ∫e^u du = e^u + C = e^(x²) + C
Common Subs tu ons:
o For √(a²-x²) or 1/(a²+x²): Try x = a sinθ or x = a tanθ.
o For e^(kx) or ln(x): Try u = kx or u = ln(x).
o For x√(x+1): Try u = x+1.
KCET Tip: If the integrand has a composite func on (e.g., sin(2x)), try u =
2x. Expect 2-3 such ques ons.
3. Definite Integrals
KCET loves definite integrals with limits (tests evalua on + proper es).
Key formulas:
Basic Evalua on: ∫(a to b) f(x) dx = F(b) - F(a), where F(x) is the
an deriva ve.
Symmetry Proper es (saves me):
o ∫(-a to a) f(x) dx = 2∫(0 to a) f(x) dx if f(x) is even (e.g., f(x) = x²,
cos(x)).
o ∫(-a to a) f(x) dx = 0 if f(x) is odd (e.g., f(x) = x³, sin(x)).
o ∫(0 to 2a) f(x) dx = 2∫(0 to a) f(x) dx if f(2a-x) = f(x).
Standard Results:
o ∫(0 to π/2) sin(x) dx = 1
o ∫(0 to π/2) cos(x) dx = 1
o ∫(0 to π/2) sin²(x) dx = π/4
KCET Tip: Check limits first. If they’re 0 to π or -a to a, use symmetry to
simplify (saves 1-2 minutes per ques on).
4. Integra on by Parts
Use when the integrand is a product (e.g., x e^x, x sin(x)).
Formula: ∫u dv = uv - ∫v du
Priority for u (LIATE rule):
o L: Logarithmic (e.g., ln(x))
o I: Inverse trig (e.g., tan⁻¹(x))
o A: Algebraic (e.g., x, x²)
o T: Trig (e.g., sin(x))
o E: Exponen al (e.g., e^x)
Example: ∫x e^x dx
o u = x, dv = e^x dx
o du = dx, v = e^x
o ∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C
KCET Tip: Expect 1-2 ques ons. Choose u as the term that
simplifies fastest (e.g., x over e^x).
5. Special Integrals
KCET occasionally throws curveballs from 2nd PUC:
Trigonometric:
o ∫sin(ax)cos(bx) dx: Use iden es like ,
sin A cos B = (1/2)[sin(A+B) + sin(A-B)].
o ∫tan(x) dx = -ln|cos(x)| + C
Ra onal Func ons:
o ∫1/(x²+a²) dx = (1/a)tan⁻¹(x/a) + C
o ∫1/√(a²-x²) dx = sin⁻¹(x/a) + C
Quadra c Denominators:
o For ∫dx/(ax²+bx+c), complete the square, then
use standard forms above.
KCET Tip: If you see trig or 1/(x²+1), don’t overthink—
apply the formula directly.
6. Prac ce Ques ons (Do These!)
Solve these to lock in concepts (answers provided for
self-check):
1. ∫(2x+3) dx = x²+3x+C
2. ∫e^(3x) dx = (1/3)e^(3x)+C
3. ∫(0 to 1) x² dx = 1/3
4. ∫sin(2x) dx = -(1/2)cos(2x)+C
5. ∫x ln(x) dx (by parts) = (x²/2)ln(x) - x²/4 + C
KCET Tip: Time yourself—aim for 1 minute per
ques on. If it takes longer, prac ce subs tu on speed.
7. Last-Minute Hacks
Memorize: Write the top 10 formulas on a s cky
note. Glance at it every morning.
Spot Pa erns: If the answer op ons are in tan⁻¹ or
sin⁻¹ form, the integral likely involves 1/(1+x²) or
1/√(1-x²).
Skip & Guess: If a ques on looks complex (e.g.,
∫sin(x)/√(1+cos²(x))), guess and move on—no
nega ve marks!
Check Units: Definite integral answers should be
numbers, not func ons. If you get x terms, you
forgot to evaluate limits.
How to Use This
Today: Copy the formulas onto a single page. Solve
the 5 prac ce ques ons above (10 minutes). Check
answers.
Tomorrow: Prac ce 10 integrals from KCET past
papers (focus on 2018-2024). Use this sheet to
spot pa erns.
Day 3: Skim the sheet before your mock test. In
the exam, recall formulas mentally before star ng
Math.
This sheet covers 90% of KCET integral ques ons
(expect 5-8 integrals in Math). If you nail these, you’re
looking at 5-6 marks guaranteed, maybe more with
luck.
