1. The problem: Create a formula sheet for Integration in class 12th mathematics. 2. Integration is the reverse process of differentiation and is used to find areas, volumes, central points, and many useful things. 3. Basic formulas: - $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ - $$\int \frac{1}{x} \, dx = \ln|x| + C$$ - $$\int e^x \, dx = e^x + C$$ - $$\int a^x \, dx = \frac{a^x}{\ln a} + C, \quad a>0, a \neq 1$$ 4. Integration of trigonometric functions: - $$\int \sin x \, dx = -\cos x + C$$ - $$\int \cos x \, dx = \sin x + C$$ - $$\int \sec^2 x \, dx = \tan x + C$$ - $$\int \csc^2 x \, dx = -\cot x + C$$ - $$\int \sec x \tan x \, dx = \sec x + C$$ - $$\int \csc x \cot x \, dx = -\csc x + C$$ 5. Integration by parts formula: - $$\int u \, dv = uv - \int v \, du$$ 6. Integration by substitution: - If $$x = g(t)$$ then $$\int f(x) \, dx = \int f(g(t)) g'(t) \, dt$$ 7. Definite integral properties: - $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $$F'(x) = f(x)$$ - Linearity: $$\int_a^b [cf(x) + dg(x)] \, dx = c\int_a^b f(x) \, dx + d\int_a^b g(x) \, dx$$ This sheet covers the essential formulas and rules for integration in class 12th mathematics.