Evaluate the integral.
cot3(4x) dx
first off, get rid of the 4x. Let
u = 4x
du = 4dx
then we have
1/4 ∫ cot^3 u du
= 1/4 ∫ cot u cot^2 u
= 1/4 ∫ cot u (csc^2 u - 1) du
= 1/4 ∫ cot u csc^2 u du - 1/4 ∫ cot u du
= -1/8 cot^2 u - 1/4 log(sin u) + C
= -1/8 cot^2(4x) - 1/4 log(sin(4x)) + C
To evaluate the integral of cot³(4x) dx, we can use the technique of substitution.
Start by letting u = 4x, then differentiate both sides with respect to x to find du/dx = 4. Solving for dx, we have dx = du/4.
Now, substitute u = 4x and dx = du/4 back into the integral:
∫cot³(4x) dx becomes ∫cot³(u) (du/4).
Next, we need to simplify the expression inside the integral. Recall that cot(x) = cos(x)/sin(x). Thus, cot³(x) can be rewritten as (cos(x)/sin(x))³ for any value of x.
In our case, x = u/4 (since u = 4x). Therefore, cot³(u) becomes (cos(u/4)/sin(u/4))³.
Now, substitute this simplified expression into the integral:
∫cot³(u) (du/4) = (1/4) ∫(cos(u/4)/sin(u/4))³ du.
To proceed, we can use the trigonometric identity: (cos(x)/sin(x))³ = cos³(x)/sin³(x).
Applying this identity to our integral, we have:
(1/4) ∫(cos(u/4)/sin(u/4))³ du
= (1/4) ∫cos³(u/4)/sin³(u/4) du.
Now, split the integral into two separate integrals:
(1/4) ∫cos³(u/4)/sin³(u/4) du
= (1/4) ∫cos²(u/4) cos(u/4)/sin³(u/4) du.
Since cos²(u/4) = (1 + cos(2u/4))/2, we can rewrite the integral as:
(1/4) ∫(1 + cos(2u/4))/2 cos(u/4)/sin³(u/4) du.
Now, simplify the integral further:
(1/8) ∫(1 + cos(u/2)) cos(u/4)/sin³(u/4) du.
Next, distribute the fractions:
(1/8) ∫((cos(u/4) + cos(u/2)cos(u/4))/(sin³(u/4))) du.
Now, split the integral into two separate integrals:
(1/8) ∫(cos(u/4)/sin³(u/4) + cos(u/2)cos(u/4)/sin³(u/4)) du.
Finally, integrate each term separately:
(1/8) ∫cos(u/4)/sin³(u/4) du + (1/8) ∫cos(u/2)cos(u/4)/sin³(u/4) du.
To evaluate these integrals further, you may need to use trigonometric identities and techniques of integration. Unfortunately, the integral of cot³(4x) does not have a simple closed-form solution, so you would typically solve it using numerical methods or computer software.