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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?

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one year 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^2so, integral J becomesJ= ∫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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one year ago
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