Tuesday, 6 August 2013

integration - Find the value of the integral: intfracpi40frac13cos2theta+sin2thetadtheta




Using the substitution t=tanθ , find the value of the integral:

π/4013cos2θ+sin2θdθ




I isolated θ in t=tanθ, and then substituted the expression into the integral. I then replaced dθ by dt, which resulted in a very ugly, large expression.



The following is the answer given:



enter image description here



I do not understand how the first integral was obtained.



Answer



Draw a right triangle with an angle θ. Since tanθ=t, you can label the side opposite θ as t, and the side adjacent to θ as 1. Then the hypotenuse is t2+1. Therefore:



cosθ=1t2+1andsinθ=tt2+1



enter image description here



Can you take it from here?


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