How do you choose u and v in integration by parts?

Use the ILATE rule for choosing u (the function to differentiate): Inverse trig > Logarithm > Algebraic > Trigonometric > Exponential. Choose u as the function higher in this priority. The other function is v (to integrate). Example: ∫x·ln x dx — u = ln x (L), v = x (A). This makes differentiation of u simpler.

What is the difference between definite and indefinite integrals?

Indefinite integral: ∫f(x)dx = F(x)+C — result is a family of functions with arbitrary constant C. No limits. Definite integral: ∫ₐᵇ f(x)dx = F(b)−F(a) — result is a specific number; no +C needed; has definite lower (a) and upper (b) limits. Geometrically represents area under f(x) from a to b.

When is integration by partial fractions used?

Integration by partial fractions is used for rational functions P(x)/Q(x) where degree of P < degree of Q. The denominator Q(x) is factored and the fraction decomposed into simpler fractions. Types of denominators: (x−a)(x−b) → A/(x−a)+B/(x−b); (x−a)² → A/(x−a)+B/(x−a)²; (x²+bx+c) irreducible → (Ax+B)/(x²+bx+c).

Integration by Parts and Integration Techniques

Integrals • Class 12 Mathematics • NCERT • CBSE

Integration by Parts: ∫u·v dx = u·∫v dx − ∫(du/dx · ∫v dx)dx. ILATE order for choosing u: Inverse trig > Logarithm > Algebraic > Trigonometric > Exponential. Key formula: ∫eˣ[f(x) + f'(x)]dx = eˣ·f(x) + C.

Key Formulas

∫u·v dx = u·∫v dx − ∫(u'·∫v dx)dx (IBP)
∫eˣ[f(x)+f'(x)]dx = eˣf(x)+C
∫xⁿ dx = xⁿ⁺¹/(n+1)+C
∫1/x dx = ln|x|+C
∫ₐᵇ f(x)dx = F(b)−F(a)

Frequently Asked Questions

How do you choose u and v in integration by parts?: Use the ILATE rule for choosing u (the function to differentiate): Inverse trig > Logarithm > Algebraic > Trigonometric > Exponential. Choose u as the function higher in this priority. The other function is v (to integrate). Example: ∫x·ln x dx — u = ln x (L), v = x (A). This makes differentiation of u simpler.
What is the difference between definite and indefinite integrals?: Indefinite integral: ∫f(x)dx = F(x)+C — result is a family of functions with arbitrary constant C. No limits. Definite integral: ∫ₐᵇ f(x)dx = F(b)−F(a) — result is a specific number; no +C needed; has definite lower (a) and upper (b) limits. Geometrically represents area under f(x) from a to b.
When is integration by partial fractions used?: Integration by partial fractions is used for rational functions P(x)/Q(x) where degree of P < degree of Q. The denominator Q(x) is factored and the fraction decomposed into simpler fractions. Types of denominators: (x−a)(x−b) → A/(x−a)+B/(x−b); (x−a)² → A/(x−a)+B/(x−a)²; (x²+bx+c) irreducible → (Ax+B)/(x²+bx+c).

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