Find the integral of ∫1/(1+tanx)dx

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Homework Statement
my teacher told me to solve this one in all the possible ways . I seem to have missed out on any one of them. Please help me oout
Relevant Equations
All standard integrals and concepts covered in AP calculus
I have done one by assuming tanx as u in substitution
 
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Rhdjfgjgj said:
Homework Statement: my teacher told me to solve this one in all the possible ways . I seem to have missed out on any one of them. Please help me oout
Relevant Equations: All standard integrals and concepts covered in AP calculus

I have done one by assuming tanx as u in substitution
Please show us. Thanks.
 
What is all possible ways? I can think of only substitution, but not the one you attempted. I think a half-angle should be better.
 
Also consider \tan(x - a) = \frac{\tan x - \tan a}{1 + \tan x \tan a}<br /> = \frac{1}{\tan a} - \left(\frac{1}{\tan a} + \tan a\right) \frac{1}{1 + \tan x \tan a}<br /> for suitable a.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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