Question

Check my work: Integraltion

Answer 1

\[\int\limits_{0}^{1}\frac{\arctan(y)}{1+y^{2}}dy\]

Answer 2

So I believe that \[\frac{1}{(1)^{2}+y^{2}}=arctan(y)?\]

Answer 3

so what it would be
\[\int \frac{arctan(y)}{arctan(y)}dy\rightarrow \int 1dy?\]

Answer 4

this isn't right

Answer 5

it's the differentiation of arctan(y) that is equal to 1/(1+ y^2)

Answer 6

yeah I messed up going to fix it

Answer 7

what that implies is substitute arctan(y) = u

Answer 8

@shubhamsrg is right. Then you need to substitute:
\[du = \frac{ 1 }{ 1 + y ^{2} } dy\]

Answer 9

\[\int\limits_{0}^{1}\frac{\arctan(y)}{1+y^{2}}dy\rightarrow \]
\[\int\limits_{0}^{1}\frac{u}{1+y^{2}} (1+y^{2})du\]

Answer 10

seems about right.