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Integrate log(1+tan x) from 0 to pi/4

Integrate log(1+tan x) from 0 to pi/4
 

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
I = \int_{0}^{\pi /4}ln(1+tanx)dx
I = \int_{0}^{a}f(x)dx = \int_{0}^{a}f(a-x)dx
I = \int_{0}^{\pi /4}ln(1+tan(\frac{\pi }{4}-x))dx
I = \int_{0}^{\pi /4}ln(1+\frac{tan\frac{\pi }{4}-tanx}{1+tan\frac{\pi }{4}.tanx})dx
I = \int_{0}^{\pi /4}ln(1+\frac{1-tanx}{1+tanx})dx
I = \int_{0}^{\pi /4}ln(\frac{2}{1+tanx})dx
I = \int_{0}^{\pi /4}ln(2)dx-\int_{0}^{\pi /4}ln(1+tan(x))dx
I = \int_{0}^{\pi /4}ln(2)dx-I
2I = \int_{0}^{\pi /4}ln(2)dx
I = \frac{ln(2)}{2}(x)_{0}^{\pi /4}
I = \frac{\pi }{8}ln(2)
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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