DEFINITE INTEGRATION
DEFINITE INTEGRAL
✤ Some property
1. ∫ₐᵇ f(x) dx. = ∫ₐᵇ f(t) ⋅ dt
2. ∫ₐᵇ f(x) dx = - ∫ₐᵃ f(x) dx
3. ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx
4. ∫₋ₐᵃ f(x) dx = 0, when, f(x) is add.
= 2∫₀ᵃ f(x) dx for even.
5. ∫ₐᵇ f(x) dx = - ∫ₐᶜ f(x) dx + ∫ₒᵇ f(x) dx
Her a < c < b
§ It is used in modulus functions.
6. ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx, f(2a-x)=f(x)
7. ∫₀ᵃ x f(x) dx = a/2 ∫₀ᵃ f(x) dx.
8. ∫₀ⁿ⁺ᵀ f(x) dx = n ∫₀ᵀ f(x) dx
If function is periodic f(T+x) = f(x).
9. ∫₀ᵃ f(x) dx = n ∫₀ᵃ/ⁿ f(x) dx
n is +ve Real number.
10. ∫ (d/dx f(x)) dx = f(x).
11. ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx.
12. ∫ₐᵇ (f(x) + g(x)) dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx.
13. ∫ₐᵇ eˣ (f(x) + f'(x)) dx = (eˣ ⋅ f(x))ₐᵇ
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