Question

NOTES
Evaluate the following
the following Integrals.
​

Answer 1

Step-by-step explanation:

We can write

sinx = sin2(x/2)

→ sinx = 2sin(x/2)cosx/2 ----- eq -- 1

and

cosx = cos2(x/2)

→ cosx = 2cos²(x/2) – 1 ----- eq -- 2

integration of tan-¹ [sinx/(1+cosx)].dx

here I will use this symbol " ! " to indicate integration

→ ! tan-¹[sinx / (1 + cosx)).dx

by replacing the value according to 1st and 2nd eq we get

→ ! tan-¹[ (2sinx/2cosx/2) / (1+2cos²x – 1].dx

→ ! yan-¹[(2sinx/2cosx/2) / ( cos²x/2)]

→ ! tan-¹[(sinx/2) / cosx/2].dx

→ ! tan-¹ [tanx/2].dx

→ ! x/2.dx

→ 1/2 [ ! x.dx ]

→ 1/2 [x²/2 + c']

→ x²/4 + c

where c = c'/2