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Question

What is the value of the integral π40ln(1+tan x)dx

A
ln (2).π8
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B
ln(2).π

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C
ln (4).π4

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D
ln (2).π3
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Solution

The correct option is A ln (2).π8

I want to say that this is a very famous question. It’s due to the fact that it emphasises the importance of the following property for its evaluation.
a0f(x)dx=a0f(ax).dx.
So understand all the properties of Definite Integrals and you can use them as an essential tool-kit to tackle this chapter.
So here, let the integral to be evaluated be `I'.
π40ln(1+tanx)dx.
=π40ln(1+tan(π4x))dx.
=π40ln(1+1tan(x)1+tan(x))dx
=π40ln(21+tan(x))dx
=π40ln(2)dxπ/40ln(1+tan(x))dx
=ln(2)x|π40l
2l=In (2).π4
I=In (2).π8

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