The integral of 1/(cos(u)^2 sin(u)) is equal to -cot(u) + C, where C is the constant of integration.
To find the integral of 1/(cos(u)^2 sin(u)), you can use the substitution method by setting cos(u) = t. This will transform the integral into -1/(t^2 - 1) which can be easily solved using partial fraction decomposition.
Step 1: Let cos(u) = t, then du = -sin(u) du
Step 2: Substitute t and du in the integral, which becomes ∫-1/(t^2 - 1) dt
Step 3: Use partial fraction decomposition to rewrite the integral as ∫-1/(t - 1) + 1/(t + 1) dt
Step 4: Integrate both terms separately to get the final answer -cot(u) + C
Yes, you can also use the trigonometric identity cos(u)^2 = 1 + tan(u)^2 to rewrite the integral as ∫1/(1 + tan(u)^2) tan(u) du. Then, use the substitution method by setting tan(u) = t to get the integral -∫1/t dt, which is easier to solve.
Yes, you can also use the u-substitution method by setting u = cos(u). This will transform the integral into -∫1/(u^2 - 1) du, which can be solved using partial fraction decomposition or by using the formula ∫1/(x^2 - a^2) dx = (1/2a)ln|(x-a)/(x+a)| + C.