📐 MATHS · CLASS 12

Integration

Area under the curve — watch the integral fill in, step by step

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∫ What is Integration?

Integration is the reverse of differentiation. It finds the area under a curve or accumulates quantities over an interval.

Indefinite Integral ∫ f(x) dx = F(x) + C where F'(x) = f(x)

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Watch the area fill in! The integral ∫₀³ x² dx = 9

∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9

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These are your tools — memorise them!

Power Rule∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Constant∫k dx = kx + C

Exponential∫eˣ dx = eˣ + C

Natural Log∫(1/x) dx = ln|x| + C

Sin∫sin x dx = −cos x + C

Cos∫cos x dx = sin x + C

Sec²∫sec²x dx = tan x + C

Cosec²∫cosec²x dx = −cot x + C

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Apply standard formulas directly. e.g. ∫3x² dx = x³ + C

Let u = g(x), find du = g'(x) dx, then integrate in terms of u.

∫u dv = uv − ∫v du Use ILATE rule to choose u.

Break rational functions into simpler fractions before integrating.

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Q: Evaluate ∫(3x² + 2x + 5) dx

= ∫3x² dx + ∫2x dx + ∫5 dx

= 3·(x³/3) + 2·(x²/2) + 5x

= x³ + x² + 5x + C

∫(3x² + 2x + 5) dx = x³ + x² + 5x + C

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Integration mastered! 🎉