Integral of (Cotx)^(1/3) divided by sinx cosx
1. Integral of (Cotx)^(1/3):
To find the integral of (Cotx)^(1/3), we can rewrite Cotx as 1/Tanx. Therefore, the integral becomes:
∫(1/Tanx)^(1/3) dx
= ∫Tanx^(-1/3) dx
= -3Tanx^(2/3) / 2 + C
2. Integral of sinx cosx:
The integral of sinx cosx can be found by using the substitution method or trigonometric identities. One way to solve it is by using the identity sin2x = 2sinxcosx:
∫sinx cosx dx
= 1/2 ∫sin2x dx
= -1/4 cos2x + C
3. Dividing the two integrals:
Now, to find the integral of (Cotx)^(1/3) divided by sinx cosx, we divide the two results obtained above:
(-3Tanx^(2/3) / 2) / (-1/4 cos2x)
= 12Tanx^(2/3) / cos2x
= 12Tanx^(2/3) / (1 - Tan^2x)
= 12Tanx^(2/3) / Sec^2x
= 12Sin^2x / Sec^(5/3)x
Therefore, the final result of the integral of (Cotx)^(1/3) divided by sinx cosx is 12Sin^2x / Sec^(5/3)x.