Integrals

Board: CBSE |Class:Class12

ComprehensivestudynotesforIntegralsbyAjayYadav.

KeyConcepts

IntegrationasAntiderivative

Integrationisthereverseprocessofdifferentiation.∫f(x)dx=F(x)+CwhereF'(x)=f(x).Cisconstantofintegration.

StandardIntegrals

∫xⁿdx=xⁿ⁺¹/(n+1)+C(n≠-1).∫1/xdx=ln|x|+C. ∫exdx = ex+C. ∫axdx = ax/lna+C.

Integration by Substitution

Let u=g(x), du=g'(x)dx. Then ∫f(g(x))g'(x)dx = ∫f(u)du. Used when integrand has a function and its derivative.

Integration by Parts

∫u dv = uv – ∫v du (ILATE rule: Inverse, Log, Algebraic, Trig, Exponential). Choose u as first function in ILATE order.

Partial Fractions

For rational functions: decompose P(x)/Q(x) into simpler fractions. Types: (x-a) gives A/(x-a). (x-a)ⁿ gives A₁/(x-a)+…+Aₙ/(x-a)ⁿ. (x²+a²) gives (Ax+B)/(x²+a²).

Definite Integrals

∫^anf(x)dx = F(b)-F(a). Properties: ∫^anf(x)dx = -∫^naf(x)dx. ∫^anf(x)dx = ∫^aef(x)dx+∫^enf(x)dx. ∫₀^af(x)dx = ∫₀^af(a-x)dx. ∫₀²f(x)dx = 2∫₀^af(x)dx if f(2a-x)=f(x).

Important Formulas

Power rule	∫xⁿdx = xⁿ⁺¹/(n+1)+C
Integration by parts	∫u dv = uv - ∫v du
Definite integral	∫anf(x)dx = F(b)-F(a)
Property	∫₀af(x)dx = ∫₀af(a-x)dx

Solved Examples

Example 1: Find ∫x³dx.

Solution: x⁴/4 + C.

Example 2: Find ∫x sinx dx using integration by parts.

Solution: u=x, dv=sinxdx. du=dx, v=-cosx. = -xcosx + ∫cosxdx = -xcosx + sinx + C.

Example 3: Evaluate ∫₀¹ x²dx.

Solution: (x&sup3/3)₀¹ = 1/3.

Practice Questions

1. ∫(3x²+2x+1)dx.
2. ∫xexdx.
3. ∫1/(x²-a²)dx.
4. ∫₀⁼/₂ sinx dx.
5. ∫₀¹ dx/(1+x²).

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