- #1

Krushnaraj Pandya

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## Homework Statement

∫1/(√3 sinx+ cosx) dx is

Ans. as per textbook 1/2(log(tan((x/2)+(pi/12))

**2. The attempt at a solution**

Attempt 1-I changed it to half angles (x/2) and then multiplied and divided sec^2(x/2) to the numerator and denominator, then putting tan(x/2)=t I got 2∫1/(√3t+1-t^2) I wrote the quadratic expression as (√7/2)^2-(t-(√3/2)^2) [completing the square], then applying the standard integral form of 1/(a^2-x^2) I substituted the values into 1/2a(ln(a+x)-ln(a-x)) but I don't know how to match it with the answer

Attempt 2- I took 1/2 common from denominator to convert it into sin(pi/3)sinx+cos(pi/3)cosx=cos(x-(pi/3))

then I integrated using ∫secx=log(secx+tanx) and ended up with 1/2(log(sec(x-(pi/3))+tan(x-(pi/3)) Again, I cant match with the answer. Would appreciate some help