Class 12 Maths: Integration Cheat Sheet

Integration is the inverse process of differentiation. Here are the most important formulas you need to memorize.

1. Basic Formulas

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
• $\int \frac{1}{x} dx = \log|x| + C$
• $\int e^x dx = e^x + C$
• $\int a^x dx = \frac{a^x}{\log a} + C$

2. Trigonometric Integrals

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• $\int \csc^2 x dx = -\cot x + C$
• $\int \sec x \tan x dx = \sec x + C$
• $\int \tan x dx = \log |\sec x| + C$

3. Special Integrals (The "Big Six")

1. $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left| \frac{x-a}{x+a} \right| + C$
2. $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log \left| \frac{a+x}{a-x} \right| + C$
3. $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C$
4. $\int \frac{dx}{\sqrt{x^2 - a^2}} = \log |x + \sqrt{x^2 - a^2}| + C$
5. $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \left( \frac{x}{a} \right) + C$
6. $\int \frac{dx}{\sqrt{x^2 + a^2}} = \log |x + \sqrt{x^2 + a^2}| + C$

4. Integration by Parts

$$\int u v dx = u \int v dx - \int \left( \frac{du}{dx} \int v dx \right) dx$$ Rule for selecting 'u': ILATE (Inverse, Log, Algebraic, Trig, Exponential).

5. Definite Integral Properties

• $\int_a^b f(x) dx = -\int_b^a f(x) dx$
• $\int_0^a f(x) dx = \int_0^a f(a-x) dx \quad \text{(King's Property)}$

Tip: Integration requires constant practice. Solve at least 10 problems daily to master these formulas! 📐