Ch 7 — the longest in the syllabus — covers both indefinite and definite integrals. Students learn integration as the reverse of differentiation, master multiple techniques, and apply the Fundamental Theorem of Calculus.
Integration is the reverse of differentiation. ∫f(x)dx = F(x) + C where F'(x) = f(x). Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1), ∫eˣdx = eˣ, ∫sin x dx = −cos x, ∫1/x dx = ln|x| + C.
Substitution: replace complex part with u. Integration by parts: ∫u·v dx = u∫v dx − ∫(du/dx · ∫v dx)dx (LIATE rule for choosing u). Partial fractions for rational functions. Special integrals for √(a²−x²), 1/(a²+x²), etc.
Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b) − F(a) where F'(x) = f(x). Properties: ∫ₐᵇ = −∫ᵇₐ, ∫ₐᵇ = ∫ₐᶜ + ∫ᶜᵇ, ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx.
Download: https://ncert.nic.in/textbook/pdf/lemh201.pdf | Complete book Part II: https://ncert.nic.in/textbook/pdf/lemh2ps.zip
It links differentiation and integration: ∫ₐᵇ f(x)dx = F(b) − F(a) where F is any antiderivative of f. This means you can evaluate definite integrals by finding antiderivatives instead of computing limits of sums.
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