# Some Standard Formulae Using Substitution

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4. = log(x + c
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 Form Working rule Divide numerator and denominator both by cos2 x, replace sec2 x, if any, in denominator by (1 + tan2 x) and put tan x = t. So that sec2 x dx = dt where m, n âˆˆ N If one of them is odd, then substitute for the term of even power If both are odd, substitute either of them If both are even, use trigonometric identities only If m and n are rational numbers and  is a negative integer, thensubstitute cot x = p or tan x = p Write sin x and cos xin terms of tan(x/2) and then substitute for tan (x/2) = t In this integral express numerator asÎ» (denominator) + Î¼(differentiation of denominator) + Î³.   Find Î», Î¼, and Î³ by comparing coefficients of sin x, cos x and constant term and split the integral into sum of three integrals. This linear factor (px+ q) is expressed in terms of the derivative of the quadratic factor ax2 +bx + c together with a constant as px + q =Î»(d/dx) {ax2 + bx +c} + Î¼ â‡’ px + q =Î»(2ax + b) + Î¼.   Here we find Î» and Î¼by comparing coefficients Substitute Substitute