definite integrals (1 Viewer)

[jkwii]

some one must have seen this integral b4.
pi/4
∫ e^x/(1+e^x)cos^2x dx
-pi/4

 

[jkwii]

had another quick query,
pi/6
∫ln(tanx + Sqrt(3)) dx
0
i think u use f(x) = f(a-x) to do it but i neek working pls.

 

[sicmacao]

had another quick query,
pi/6
∫ln(tanx + Sqrt(3)) dx
0
i think u use f(x) = f(a-x) to do it but i neek working pls.

Using the following fact
pi/6 pi/6
∫f(x)dx = ∫f(pi/6-x)dx
0 0

pi/6
let I = ∫ln(tanx + Sqrt(3)) dx
0

pi/6
∫ln(tanx + Sqrt(3)) dx =
0

pi/6
∫ln(tan(pi/6-x) + Sqrt(3)) dx =
0

pi/6
∫ln( (1- sqrt(3)tanx)/(sqrt(3)+tanx) + Sqrt(3)) dx =
0

(common denom inside ln)

pi/6
∫ ln(4/(sqrt(3)+tanx)) dx =
0

pi/6
∫ ln4 - ln(sqrt(3)+tanx)dx =
0

pi/6
∫ ln4 - I
0

Therefore 2I = (pi*ln4)/6

hence I=(pi*ln4)/12

Q.E.D.

This is an interesting question to do. Which book did you find that?

 

[jkwii]

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