Integration and Definite Integrals

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Integration — Key Facts

Integration is the reverse process of differentiation. If F’(x) = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration.

Basic Integrals:

Function	Integral
$x^n$	$\frac{x^{n+1}}{n+1} + C$ (n ≠ -1)
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$e^x$	$e^x + C$
$\frac{1}{x}$	$\ln
$a^x$	$\frac{a^x}{\ln a} + C$

⚡ ECAT Exam Tip: Don’t forget +C (constant of integration) for indefinite integrals! For definite integrals, the limits take the place of +C.

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Methods of Integration:

1. Substitution:

When integral is of form ∫f(g(x)) · g’(x) dx, let u = g(x) $$\int f(g(x)) \cdot g’(x) dx = \int f(u) du$$

Example: ∫sin³x cosx dx Let u = sinx, then du = cosx dx ∫u³ du = u⁴/4 + C = sin⁴x/4 + C

2. Integration by Parts:

$$\int u dv = uv - \int v du$$

LIATE Rule (choose u in this order):

• Logarithmic
• Inverse trigonometric
• Algebraic
• Trigonometric
• Exponential

Example: ∫x eˣ dx Let u = x, dv = eˣdx → du = dx, v = eˣ ∫x eˣ dx = x eˣ - ∫eˣ dx = x eˣ - eˣ + C = eˣ(x-1) + C

3. Partial Fractions:

For rational functions where degree of numerator < degree of denominator:

Case 1: Linear factors $$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$

Case 2: Repeated linear factors $$\frac{1}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$$

Definite Integrals:

$$\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)$$

⚡ ECAT Exam Tip: When using substitution in definite integrals, change the limits (a and b) to u-values. Don’t forget this — many students forget and get wrong answers.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Advanced Integration Techniques:

1. Trigonometric Substitutions:

Expression	Substitution	Identity
$\sqrt{a^2-x^2}$	x = a sinθ	$1 - \sin^2\theta = \cos^2\theta$
$\sqrt{a^2+x^2}$	x = a tanθ	$1 + \tan^2\theta = \sec^2\theta$
$\sqrt{x^2-a^2}$	x = a secθ	$\sec^2\theta - 1 = \tan^2\theta$

2. Integration of Special Functions:

$$\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\frac{x}{a} + C$$ $$\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$$ $$\int \frac{1}{x\sqrt{x^2-a^2}} dx = \frac{1}{a}\sec^{-1}\frac{x}{a} + C$$

3. Reduction Formulas:

For integrals of powers of trig functions: $$\int \sin^n x dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2}x dx$$

$$\int \cos^n x dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2}x dx$$

4. Integration of Rational Functions by Partial Fractions:

Case 3: Irreducible quadratic factors $$\frac{px+q}{(x-a)(x^2+bx+c)} = \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$$

Case 4: Repeated quadratic factors $$\frac{px+q}{(x^2+bx+c)^2} = \frac{Ax+B}{x^2+bx+c} + \frac{Cx+D}{(x^2+bx+c)^2}$$

5. Area Under Curves:

Area between curve y = f(x) and x-axis from a to b: $$A = \int_a^b |f(x)| dx$$

Area between two curves: $$A = \int_a^b |f(x) - g(x)| dx$$

6. Volume of Revolution:

Using disc method: $$V = \pi \int_a^b [f(x)]^2 dx \quad \text{(rotate around x-axis)}$$ $$V = \pi \int_a^b [f(y)]^2 dy \quad \text{(rotate around y-axis)}$$

Using shell method (rotate around y-axis): $$V = 2\pi \int_a^b x f(x) dx$$

7. Improper Integrals:

When limits are infinite or function is discontinuous:

$$\int_a^\infty f(x) dx = \lim_{b\to\infty} \int_a^b f(x) dx$$

$$\int_{-\infty}^\infty f(x) dx = \lim_{a\to-\infty}\lim_{b\to\infty}\int_a^b f(x) dx$$

8. Differential Equations:

First order separable: $$\frac{dy}{dx} = f(x)g(y) \Rightarrow \int \frac{dy}{g(y)} = \int f(x) dx$$

Linear first order: $$\frac{dy}{dx} + Py = Q \Rightarrow \text{Integrating Factor} = e^{\int P dx}$$

⚡ ECAT 2024 Analysis: Questions on integration by parts (LIATE rule) and area under curves are frequently tested. For definite integrals with absolute values, always identify where f(x) = 0 to split the integral. The volume of revolution is also in the ECAT syllabus.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.