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Working rules of Integration |
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[f(x) + g(x)] dx = ?f(x) dx + ?g(x) dx |
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?k*f(x)
dx = k * ?f(x) dx
( k = constant) |
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u*v dx = u?v dx - ?
[(du/dx)?v dx] dx |
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f(x) dx = [ F(x) ]ab = F(b) - F(a) |
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Integration formula |
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xn dx = x n+1 / n + 1 |
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?n dx = n*x + c
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1/x dx = log |x| + c |
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cos x dx = sin x + c |
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sin x dx = - cos x + c |
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sec2x dx = tan x + c |
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cosec2x dx = - cot x + c |
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sec x * tan x dx = sec x + c |
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cosec x * cot x dx = - cosec x + c |
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dx / (x2 + a2) = (1/a * tan-1 x/a + c) = (- 1/a * cot-1 x/a + c1) |
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dx / sqrt(a2 - x2) = (sin-1 x/a + c) = (-cot-1 x/a + c1) |
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dx / (|x| * sqrt(x2 - a2)) = (1/a * sec-1 x/a + c) = (-1/a * cosec-1 x/a
+ c1) |
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ax dx = (ax / logea) + c |
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ex dx = ex + c |
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dx / sqrt(x2 +/- a2) = log |x + sqrt(x2 +/- a2)| + c |
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dx / (x2 - a2)= ((1/(2*a)) * log | (x-a) / (x+a) | + c |
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dx / (a2 - x2)= ((1/(2*a)) * log | (x+a) / (x-a) | + c |
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tan x dx = log |sec x| + c |
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cot x dx = log |sin x| + c |
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cosec dx = log |tan x/2| + c = log |cosec x - cot x| + c |
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sec dx = log |tan (?/4 + x/2)| + c = log
|sec x + tan x| + c |
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?sqrt(x2 + a2)
dx = [(x/2) * sqrt(x2 + a2)] + [(a2/2) *
log |x + sqrt(x2 + a2)|] + c |
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?sqrt(x2
- a2) dx = [(x/2) * sqrt(x2 - a2)] - [(a2/2)
* log |x + sqrt(x2 - a2)|] + c |
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?sqrt(a2
-x2) dx = [(x/2) * sqrt(a2-x2)]
+ [(a2/2) * sin-1 (x/a)] + c |