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UniversityCalculus
Integration (Calculus)
Finding areas and reversing differentiation
Master integral calculus — finding areas under curves and solving differential equations.
✓ ∫xⁿ dx = xⁿ⁺¹/(n+1) + C✓ Always add + C for indefinite integrals✓ Definite integral: F(b) − F(a)
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📖 Understanding Integration (Calculus)
Integration is the reverse of differentiation. The indefinite integral ∫f(x)dx gives a family of functions F(x) + C, where C is the constant of integration.
Power Rule for Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). Increase the power by 1 and divide by the new power.
Definite integrals ∫ₐᵇ f(x)dx calculate the exact area under a curve between x=a and x=b. The result is F(b) − F(a) (no constant C needed).
Key integrals: ∫eˣ dx = eˣ + C, ∫(1/x)dx = ln|x| + C, ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C.
🔑 Key Points to Remember
- ✓∫xⁿ dx = xⁿ⁺¹/(n+1) + C
- ✓Always add + C for indefinite integrals
- ✓Definite integral: F(b) − F(a)
- ✓∫eˣ dx = eˣ + C
- ✓Area between curves: ∫[f(x)−g(x)]dx
#integration#calculus#area#university#integral
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