Help me, this one is an trigono integer

[MethMath11]

f_n(x) = sin^n x
And
The question is , the value of X,Y, and Z

 

[MarkFL]

If we use the given \(f_n(x)\) and apply the double angle identity for sine, we may write:

[MATH]I_n=2\int_0^{\frac{\pi}{2}} \sin^{n+1}(x)\cos(x)\,dx[/MATH]
What do you get when you use the substitution \(u=\sin(x)\) ?

 

[MethMath11]

If we use the given \(f_n(x)\) and apply the double angle identity for sine, we may write:

[MATH]I_n=2\int_0^{\frac{\pi}{2}} \sin^{n+1}(x)\cos(x)\,dx[/MATH]
What do you get when you use the substitution \(u=\sin(x)\) ?

How to do the limit? Got any solution for it?

 

[MarkFL]

Why don't we work through the problem from start to finish instead of jumping straight to the end. It usually works better that way. I haven't even looked at the limit yet, because it depends on previous work which hasn't been done yet.

Please answer the question I posted, and rewrite the definite integral in terms of \(u\).

 

[MethMath11]

Why don't we work through the problem from start to finish instead of jumping straight to the end. It usually works better that way. I haven't even looked at the limit yet, because it depends on previous work which hasn't been done yet.

Please answer the question I posted, and rewrite the definite integral in terms of \(u\).

already got my answer without integraling it, thanks for the help and sorry for my limited vocabuy

 

[MarkFL]

I get:

[MATH]I_n=2\int_0^1 u^{n+1}\,du=\frac{2}{n+2}\left[u^{n+2}\right]_0^1=\frac{2}{n+2}[/MATH]
Now, using:

[MATH]\int_a^b f(x)\,dx=\lim_{n\to\infty}\left(\frac{b-a}{n}\sum_{k=1}^{n}f\left(x_k\right)\right)[/MATH]
We may write:

[MATH]\lim_{n\to\infty}\sum_{k=n-1}^{2(n-1)}I_k=\int_0^W \frac{X}{Y+x}\,dx[/MATH]
[MATH]\lim_{n\to\infty}\sum_{k=1}^{n}\left(\frac{2}{k+n}\right)=\int_0^W \frac{X}{Y+x}\,dx[/MATH]
[MATH]\lim_{n\to\infty}\left(\frac{1}{n}\sum_{k=1}^{n}\left(\frac{2}{\dfrac{k}{n}+1}\right)\right)=\int_0^1 \frac{2}{1+x}\,dx=2\int_1^{2}\frac{1}{u}\,du=\log(4)[/MATH]