Chapter 7 Integrals

Notes

Notes:

Integrals chapter is one of the most important chapters of Class 12 Mathematics.

Integration means reverse process of differentiation.

If d/dx F(x) = f(x), then ∫ f(x) dx = F(x) + C

Here C is called constant of integration.

Main topics of this chapter:

• Indefinite integrals • Basic integration formulas • Integration by substitution • Integration by parts • Integration using partial fractions • Definite integrals • Properties of definite integrals

Important Basic Formulas

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ –1

∫ 1/x dx = log |x| + C

∫ eˣ dx = eˣ + C

∫ aˣ dx = aˣ/log a + C

∫ sin x dx = –cos x + C

∫ cos x dx = sin x + C

∫ sec²x dx = tan x + C

∫ cosec²x dx = –cot x + C

∫ sec x tan x dx = sec x + C

∫ cosec x cot x dx = –cosec x + C

Important Inverse Trigonometric Integration Formulas

∫ 1/(1 + x²) dx = tan⁻¹x + C

∫ 1/√(1 – x²) dx = sin⁻¹x + C

∫ 1/(a² + x²) dx = 1/a tan⁻¹(x/a) + C

∫ 1/√(a² – x²) dx = sin⁻¹(x/a) + C

∫ 1/(x² – a²) dx = 1/(2a) log |(x – a)/(x + a)| + C

∫ 1/(a² – x²) dx = 1/(2a) log |(a + x)/(a – x)| + C

Integration by Substitution

Use this method when function is in composite form.

Example: ∫ 2x cos(x²) dx

Take t = x² Then dt = 2x dx

So answer = sin(x²) + C

Shortcut: If derivative of inside function is present outside, use substitution.

Integration by Parts

Formula: ∫ u v dx = u∫v dx – ∫[d(u)/dx · ∫v dx] dx

Use ILATE rule to choose first function.

ILATE means: I = Inverse trigonometric L = Logarithmic A = Algebraic T =…