by barnamah » Wed 06 April, 2011 08:09
to integrate `int4z/(1+z^4)dz`
now at the denominator we have `1+z^4` we know `z^4=(z^2)^2` so the denominator can be written as `1+(z^2)^2` so let's wire it in this format
`int4z/(1+(z^2)^2)dz`
let `u=z^2`
`du=2zdz` divide both side by 2 and we have:
`(du)/2=zdz`
now sub in the values and we got:
`int(4/((1+u^2))*1/2)du`
which we write as `int(1/((1+u^2))*4/2)du` and simplify to `int(1/((1+u^2))*2)du`
because 2 is constant, we can move it outside integral and we have:
`2int(1/(1+u^2))du`
Now it is simple. We know that `d/dxarctanx=1/(1+x^2)`
so if `1/(1+x^2)` is the derivative of `arctanx` we can use it here
`2int(1/(1+u^2))du=2(arctan(u))+C` where C is a constant.
we had `u=z^2` so sub it in and we have `2(arctan(z^2))+C`
Explore and know. That is asked.