Integration





Definition of integration:

A function g(x) is called am anti derivative or integral of a function g(x) on an interval I. If g'(x) = g(x) for every value of x in I.

If the derivative of a function G(x) with respect to x is g(x) then we can say that integral of g(x) with respect to x is g(x). Now we are going to see formulas in this topic.

Formulas:

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c


∫ (1/xⁿ) dx = -1/(n - 1) x⁽ⁿ⁻¹⁾  + c


∫ (1/x) dx = log x  + c


∫ e^(x) dx = e^x  + c


∫ a^(x) dx = a^x/(log a)  + c


∫ Sin x dx = - Cos x + c


∫ Cos x dx = Sin x + c


∫ Cosec ² x dx = - Cot x  + c


∫ Sec ² x dx = tan x  + c


∫ sec x tan x dx = sec x  + c


∫ Cosec x cot x dx = - Cosec x  + c


∫ 1/(1 + x ²) dx =  tan ⁻ ¹x  + c


∫ 1/(1 - x ²) dx = Sin ⁻ ¹x   + c


∫  e^(a x + b) =  (1/a) e^ (a x + b) + C


∫ (ax + b) dx = (1/a) (ax + b)⁽ⁿ ⁺ ¹⁾/(n + 1)   + c


1/(ax + b) dx = (1/a) log (ax + b) + c


e^(ax + b) dx = (1/a) e^ (ax + b) + c


∫ Sin (ax + b) dx = -(1/a) Cos (ax + b) + c


∫ Cos (ax + b) dx = (1/a) Sin (ax + b) + c


∫ Sec ² (ax + b) dx = (1/a) tan (ax + b) + c


∫ Cosec ² (ax + b) dx = -(1/a) cot (ax + b) + c


∫ Cosec (ax+b)cot (ax+b)dx=-(1/a)Cosec (ax+b) + c


∫ sec (ax + b) tan (ax + b) dx = sec (ax + b) + c


∫ 1/1+ (ax) ² dx = (1/a) tan ⁻ ¹ (ax) + c


∫ 1/[1 - (ax ²)] dx = (1/a) Sin ⁻ ¹(ax)   + c

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