Final answer:
Using substitution and using trigonometric identities, we can rewrite the integral and integrate each term separately to find the final solution as (2x⁵ - 5x) - cot(2x⁵ - 5x) + C.
Step-by-step explanation:
To evaluate the given integral: ∫ (10-20x⁴)csc²(2x⁵-5x) dx
We can use substitution to solve this integral.
Let u = 2x⁵ - 5x, then differentiate u with respect to x to find du/dx = 10x⁴ - 5.
Now, we can rewrite the integral in terms of u:
∫(10-20x⁴)csc²(2x⁵-5x) dx
= ∫csc²(u) du.
Using a trigonometric identity, csc²(u) = 1 + cot²(u), we can rewrite the integral as ∫(1 + cot²(u)) du.
Integrating each term separately, we get:
∫1 du = u + C1
∫cot²(u) du = -cot(u) + C2
Therefore, the final solution is:
∫(10-20x⁴)csc²(2x⁵-5x) dx
= u - cot(u) + C
Substituting u back in terms of x, we have:
∫(10-20x⁴)csc²(2x⁵-5x) dx
= (2x⁵ - 5x) - cot(2x⁵ - 5x) + C