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integrate the following ∫dx/(5 +7 cos x+sin x) the answer is 1/5 log|tan x/2+2|/|tan x/2-2| + c but how please solve?

integrate the following
∫dx/(5 +7 cos x+sin x)
 
the answer is 
1/5 log|tan x/2+2|/|tan x/2-2| + c
 
but how please solve?

Grade:11

1 Answers

Aditya Gupta
2081 Points
4 years ago
hello apurva. this is an ez ques.
we need to use the following standard formulae: sinx= 2t/(1+t^2) and cosx= (1 – t^2)/(1+t^2) where t= tan(x/2).
so we substitite t= tan(x/2) so that dt/dx= ½ *sec^2(x/2) or 2dt/dx= 1+ tan^2(x/2)= 1+t^2
so, integral J becomes
J= ∫dx/(5 +7 cos x+sin x)= ∫2dt/[5(1+t^2)+7(1 – t^2)+2t]
= 2∫dt/( –2t^2 + 2t + 12)
= ∫dt/( –t^2 + t + 6)
= – ∫dt/(t – 3)(t+2)
now you can easily use partial fractions to write 1/(t – 3)(t+2)= [1/(t – 3) – 1/(t+2)]/5 and solve it.
kindly approve :)

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