Question

pls, help to compute

Answer 1

pls help me to compute this indefinite integral
\[\int\limits_{}^{}dx/(1+x ^{2})\]

Answer 2

It is \[\tan^{-1} (x)\]. To understand why, lets calculate the derivative of \[y=\tan^{-1} (x)\].
So here, we cannot do anything, but if we apply inverse function
\[\tan (y)=x\]
Now we can derivate both sides using implicit differentation:
\[d/dx(\tan (y))=d/dx(x)\]\[y'.\sec ^{2}(y)=1\]\[y'=1/\sec^{2}(y)\]
Now I use the identify \[\sec^{2}=1+\tan^{2}\] and \[tan^{2}(y)=x^2\] to get the answer \[y'=1/(1+tan^{2}(y))=1/(1+x^{2})\]

If you cant' remember this, you can always solve the integral by trig substitution making \[x=\tan(\theta)\]. You will end up with the following integral: \[\int_{}^{}d \theta\] which is \[\theta\], since \[\tan(\theta)=x\] we then have \[\theta=\tan^{-1}(x)\].

One last thing is that a omitted the added constant from the results of the integrals just to make things easier to understand, but remember to include it in your answers.

Answer 3

thanks a lot walac, I was abel to solve that after I read about inverse functions. Any way, many thanks for your detailed reply

Answer 4

You are welcome.