Integration – Study Notes for TS EAMCET (Mathematics)
Goal: Master the concepts, techniques, and common integrals needed for the integration section of the TS EAMCET.
1. Simple Explanation
- Integration is the reverse operation of differentiation.
- It gives the area under a curve, the accumulated quantity, or the antiderivative of a function.
- Indefinite Integral (∫ f(x) dx) = a family of functions whose derivative is f(x).
- Definite Integral (∫ₐᵇ f(x) dx) = the net signed area between f(x) and the x‑axis from x = a to x = b.
2. Key Points
| Concept | What to Know | Typical EAMCET Usage |
|---|---|---|
| Fundamental Theorem of Calculus (FTC) | 1) If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) – F(a). 2) d/dx (∫ₐˣ f(t) dt) = f(x). | Computing definite integrals quickly; verifying antiderivatives. |
| Basic Antiderivatives | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1) ∫eˣ dx = eˣ + C ∫sin x dx = –cos x + C ∫cos x dx = sin x + C ∫1/x dx = ln |
x |
| Integration Techniques | 1. Substitution 2. Integration by Parts 3. Partial Fractions 4. Trigonometric & Trigonometric Substitution 5. Inverse Trig & Hyperbolic |
Most EAMCET problems use at least one of these. |
| Common Integral Forms | • ∫sinⁿ x dx, ∫cosⁿ x dx • ∫sec x tan x dx, ∫csc x cot x dx • ∫tan x dx, ∫sec x dx • ∫(ax + b)ⁿ dx • ∫1/√(a² – x²) dx, ∫1/√(x² + a²) dx • ∫1/(x² + a²) dx |
Memorise a few; rest can be derived. |
| Definite Integral Properties | • ∫ₐᵇ f(x) dx = –∫ᵦₐ f(x) dx • ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx • If f is even: ∫₋ᵃᵃ f(x) dx = 2∫₀ᵃ f(x) dx • If f is odd: ∫₋ᵃᵃ f(x) dx = 0 |
Exploiting symmetry reduces calculation time. |
| Improper Integrals | Treat limits → ∞ or integrand → ∞ as limits. | Rarely appear but good to know for completeness. |
| Notation | • ( \displaystyle \int_a^b f(x),dx ) – definite integral • ( \displaystyle \int f(x),dx ) – indefinite integral • ( \displaystyle \int_{a}^{b} f(x),dx = \lim_{n\to\infty}\sum f(x_i^*)\Delta x ) – Riemann sum interpretation |
Helps in checking if a problem is asking for an area. |
3. Worked Example
Problem (Typical TS EAMCET)
Evaluate the definite integral:
[ I = \int_{0}^{\frac{\pi}{4}} \frac{\tan x}{1+\tan x},dx ]
Step‑by‑Step Solution
- Simplify the integrand
Let ( t = \tan x ).
Then ( dt = \sec^2 x,dx = (1+\tan^2 x),dx = (1+t^2)\