Basic Integration
| Power Rule |
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| (i) ∫ 1 dx = x + C | ∫ xn dx = x^{n+1}\over n+1 + C |
| Exponential Function |
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| (ii) ∫ex dx = ex + C | ∫emx dx = 1\over memx + C |
| (iii) ∫ax dx = ax ln a + C | ∫amx dx = 1\over mamx ln a + C |
| (iv) ∫1\over x dx = ln |x| + C | ∫1\over mx dx = ln |x| + C |
| Trigonometric Functions | |
| (iii) ∫sin(x) dx = -cos(x) + C | ∫sin(mx) dx = – 1\over mcos(mx) + C |
| (iv) ∫cos(x) dx = sin(x) + C | ∫cos(mx) dx = 1\over m sin(mx) + C |
| (v) ∫sec(x)tan(x) dx = sec(x) + C | ∫sec(mx)tan(mx) dx = 1\over msec(mx) + C |
| (vi) ∫sec2(x) dx = tan(x) + C | ∫sec2(mx) dx = 1\over mtan(mx) + C |
| (vii) ∫cosec2(x) dx = -cot(x) + C | ∫cosec2(mx) dx = –1\over mcot(mx) + C |
| (viii) ∫cosec(x)cot(x) dx = -cosec(x) + C | ∫cosec(mx)cot(mx) dx = –1\over mcosec(mx) + C |
| (ix) ∫tan(x) dx = -ln|cos x| + C | |
| (x) ∫cot(x) dx = ln|sin x| + C | |
| (xi) ∫cot(x) dx = ln|sin x| + C | |
| (xii) ∫sec(x) dx = ln|sec x + tan x| + C | |
| (xiii) ∫cosec(x) dx = ln|cosec x – cot x| + C | |
Inverse and Log Integration
| (i) ∫1\over \sqrt{a^2-x^2} dx = sin-1 x\over a + C |
| (ii) ∫{1\over {a^2 +x^2} } dx = 1\over a tan-1 x\over a + C |
| (iii) ∫1\over x \sqrt{x^2-a^2} dx = 1\over a sec-1 x\over a + C |
| (iv) ∫1\over \sqrt{x^2+a^2} dx = ln |x + \sqrt{x^2+a^2}| |
| (v) ∫1\over \sqrt{x^2-a^2} dx = ln |x + \sqrt{x^2-a^2}| |
| (vi) ∫\sqrt{a^2-x^2} dx = x\sqrt{a^2-x^2}\over 2 + a^2\over 2sin-1x\over a + C |
| (vii) ∫\sqrt{a^2+x^2} dx = x\sqrt{a^2+x^2}\over 2 + a^2\over 2ln |x + \sqrt{a^2+x^2}| |
| (viii) ∫\sqrt{x^2-a^2} dx = x\sqrt{x^2-a^2}\over 2 + a^2\over 2ln |x + \sqrt{x^2-a^2}| |
| (ix) ∫1\over x^2-a^2 dx = {1\over 2a} ln |{x-a\over x+a}| |
| (x) ∫1\over a^2-x^2 dx = {1\over 2a} ln |{a+x\over a-x}| |
Integration by part
∫uv dx = u∫vdx – ∫[du/dx∫vdx]dx
U and V is selected on the basis of ILATE
Where
- I → Inverse trigonometry
- L → Logarithmic function
- A → Algebraic function
- T → Trigonometric function
- E → Exponential
Some Special Integration
- ∫ {f ‘(x)/f (x) } dx = ln f(x) + C
- ∫ex [f(x) + f ‘(x)] dx = ex f(x) + C
Partial Fraction
Definite Integrals
\int_a^b f(x) dx = [F(x)]_a^b = F(b) – F(a)
Properties of Definite Integrals
Summation of Series by Definite Integral
Let f(x) be a continuous function in [a, b] and h be the length of n equal subintervals, then
\int_{a}^{b} \ f(x) \, dx = \lim_{n \to \infty } h \sum_{r=0}^{n}{f(a+rh)}where, nh = b – a
Now, put a = 0, b = 1
∴ nh = 1 – 0 = 1 or h = 1/n
∴ \int_{0}^{1} \ f(x) \, dx = \lim_{n \to \infty } {1\over n} \sum_{r=0}^{n-1}{f(r/n)}
