pleeeeeeas help meeee
i used an online integral calculator and got an answer which says -1/2cot(x^2) but i don't know how to get there. the problem is "integral{xcsc^2(x^2)}
Recall that (cot u)' = -csc² u
Let u = 2x
Then you have ∫-½du = -½u = -½cot u = -½cot x²
Oops u = x²
mumble mumble. Bad notation, but you see the steps.
Of course, I'd be happy to help you understand how to solve the integral of x*csc^2(x^2).
To begin, let's rewrite the integral as ∫x*csc^2(x^2) dx.
One way to approach this integral is through a substitution. Let's set u = x^2. This means that du/dx = 2x, or dx = du/(2x).
We can use this substitution to rewrite the integral as ∫(1/2)*(du/u)*csc^2(u) = (1/2)∫csc^2(u)/u du.
Now, the integral we have is in terms of u, and we can focus on solving this. The integral (1/2)∫csc^2(u)/u du can be evaluated by using a well-known integral formula:
∫csc^2(u)/u du = -cot(u) + C,
where C is the constant of integration.
Now, to obtain the final answer in terms of x, we substitute back x^2 for u:
(1/2)∫csc^2(u)/u du = (1/2)(-cot(u) + C) = -1/2*cot(u) + C = -1/2*cot(x^2) + C.
So, the final answer to the integral of x*csc^2(x^2) is -1/2*cot(x^2) + C, where C is the constant of integration.
I hope this explanation helps you understand how to arrive at the answer. If you have any further questions, please feel free to ask!