Question

Integration x.tanx.sec^2x

Answer 1

∫tanxsec2xdx

Let u=cosx

1

u(1−2

u

2

)

⟹A(1−2

u

2

)+u(Bu+C)

⟹(−2A+B)

u

2

+Cu+A

Comparing coefficients...

A=1,

−2A+B

B

=∫

sinx

cosx

⋅

1

2

cos

2

x−1

dx

⟹du=−sinxdx

=∫

−du

u(2

u

2

−1)

=∫

du

u(1−2

u

2

)

=

A

u

+

Bu+C

1−2

u

2

=1

=1

C=0

=0

=2A=2

=∫

1

u

+

2u

1−2

u

2

du

=ln|u|−

1

2

ln|1−2

u

2

|+C

=

ln|cosx|−

1

2

ln|1−2

cos

2

x|+C