Question

integration

Answer 1

\[\int\limits_{}^{}\sqrt{\tan x} ~~dx\]

Answer 2

Make a substitution \[t^2 = tanx\]

Answer 3

\[x = \tan^{-1} t^2\] can you do the rest?

Answer 4

i found\[\int\limits_{}^{}\frac{ 2t^2 }{ 1+t^4 }dt\]
now?

Answer 5

\[\int\limits \frac{ 2t^2 }{ 1+t^4 }dt \implies \int\limits \frac{ t^2+1+t^2-1 }{ t^4+1 } dt = \int\limits \frac{ t^2+1 }{ t^4+1 }dt + \int\limits \frac{ t^2-1 }{ t^4+1 }\]

Answer 6

you'll have to do the integrals separately now

Answer 7

how?

Answer 8

\[\int\limits \frac{ t^2+1 }{ t^2+1 } dt \implies \int\limits \frac{ 1+1/t^2 }{ t^2+1/t^2 } dt\] now do another u substation do so similarly with other integral.

Answer 9

\[\int\limits_{}^{}\frac{ 1-1/t^2 }{t^2+1/t^2 }\]

Answer 10

Let u = t-1/t^2 for the first integral and similarly you would let u = t+1/t for the second integral (the one you presented)