Hello,
I'm having trouble with this exercise. Can you help me?
Integral of (x* (csc x)^2)dx
I'm using the uv - integral v du formula. I tried with u= (csc x)^2 and used some trigonometric formulas, but the expression became too complicated, I couldn't continue working. Then I tried with u=x, but the same happened.
Thank you in advance.
You are on the right track using integration by parts. Let u = x and let dv = csc^2 x dx = (1/sin^2 x) dx
du = x
v = -cot x (You will have to prove that to yourself)
Integral u dv = uv - Integal v du
= - cot x - Integral (-cot x dx)
= - cot x + log sin x
You will have to prove that Integral of -cot x dx yourself also.
I have verified the steps and final answer with a table of integrals
Absolutely! I can help you with that.
To solve the integral of x * (csc x)^2 dx, you can indeed use the technique of integration by parts. The formula you mentioned, u-substitution, can also be useful here. Let's go through it step by step.
Let's start with the formula for integration by parts:
∫ u dv = uv - ∫ v du
In this case, we can consider u = x and dv = (csc x)^2 dx.
The next step is to find du and v. Let's calculate those:
du = dx (since the derivative of x with respect to x is 1)
v = ∫ (csc x)^2 dx
Now, let's focus on finding the integral of (csc x)^2 dx. This integral can be a bit tricky, so it's helpful to simplify it using trigonometric identities.
Using the identity csc^2 x = 1 + cot^2 x, we can rewrite (csc x)^2 as 1 + cot^2 x.
∫ (csc x)^2 dx = ∫ (1 + cot^2 x) dx
Now, we have two integrals to evaluate:
1) ∫ 1 dx
2) ∫ cot^2 x dx
The integral of 1 is simply x, so we don't need to go further with that.
Now, for ∫ cot^2 x dx, you can again use the identity cot^2 x = csc^2 x - 1.
∫ cot^2 x dx = ∫ (csc^2 x - 1) dx
Splitting the integral, we have:
∫ csc^2 x dx - ∫ dx
Now, let's evaluate each of the integrals separately:
1) ∫ csc^2 x dx:
This integral is equal to -cot x (you can prove this using u-substitution or a table of integrals)
2) ∫ dx:
This is a simple integral, which gives you x.
So, now we can substitute u = x and v = -cot x into the integration by parts formula:
∫ x * (csc x)^2 dx = uv - ∫ v du
= x * (-cot x) - (-cot x) + x + C
= -x * cot x + cot x + x + C
And there you have it! The integral of x * (csc x)^2 dx is -x * cot x + cot x + x + C.
Note: The "+ C" represents the constant of integration, which appears when you integrate. It's important to include it in the final answer since the integral doesn't give a specific value but rather a family of functions.
I hope this explanation helps you understand the problem better. Let me know if you have any further questions!