I wish to draw your attention to your reply dt. Nov 3 to my Q 2 regarding integral of dx/(4-5 sin x). I reproduce an extract below-
"(2)...attempt a completing the square to make (2t^2-5t+2) be equal to 2(t-5/4)^2-9/8,then try a w-substitution with w = t-5/4 and you will be having integral of dw/(2w^2 -9/8). Now bring out a factor of (-2) from the denominator, and you have a similar integral...."

I tried and repeatedly got 1/3*log[2(t-2)/2t-1)] instead of 1/3*log[(t-2)/2t-1)] that I get from Partial Fractions and is given as correct answer.    (t=tan x/2)

s=1/2*Int dw/(w^2- 9/16=1/2*1/2*4/3*log[(w- 3/4)/(w+ 3/4)]

I assume that this is =1/3*log[(t-2)/2t-1)]+log 2, and the second term is accounted for by constant of integration.

Kindly advise if I am correct or whether I am making a mistake somewhere.

Yes,  both answers 1/3*log[2(t-2)/2t-1)]  and  1/3*log[(t-2)/2t-1)]  are correct.  To know whether F(x) is an integral of f(x), simply check if F'(x) is f(x).  Since both answers differ only by a constant (and the derivative of a constant is zero), both answers are correct.

Sometimes, it is not so easy to see that TWO different answers differ only by a constant, for example, when integrating sec^2(3x) tan^3(3x) dx,
one could do u = tan (3x) (du = 3sec^2(3x) dx ) or u = sec(3x)  (then du=3 sec(3x) tan(3x) dx ) and get two very different answer.  (Try it, since you know many techniques of  integration, this may further help you in sharpening your Math skills.  ) Unless one is extremely good with trigonometry, one may not even know what constant the two different answers differ by.  


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