Question

Solve this sum by integration: integration of tan inverse(cosec x+cot x)dx=?

Answer 1

Hi Please find the attachment
Hope this helps.

Answer 2

∫tan-¹(cosecx +cotx)dx = (π/2)x-(x²/4)+C

Explanation:

•Given expression can be written as

∫tan-¹(cosecx+cotx)dx

•tan-¹(cosecx+cotx) can be written as

Tan-¹((1/sinx)+(cosx/sinx))

• Now, it will be by solving,

Tan-¹((1+cosx)/sinx)

•(1+cosx) can be written as 2cos²(x/2) and sinx can be written as (2sin(x/2)cos(x/2))

•Therefore,tan-¹((1+cosx)/sinx) = tan-¹((2cos²(x/2))/2sin(x/2)cos(x/2))

•Then it can be written as

Tan-¹(cot(x/2)) = tan-¹(tan((π/2)-(x/2)))

•hence,it is (π/2)-(x/2)

•Now,we have to find integral of ((π/2)-(x/2))

•therefore,

I = ∫((π/2)-(x/2))dx

I = (π/2)x-(x²/4)+C.