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## Homework Statement

Given the function in x

$$f_n(x)=sin^nx (n=1,2,3,...)$$

For this ##f_n(x)##, consider the definite intergral

$$I_n=\int_{0}^{\pi/2}f_n(x)sin2xdx$$

a) Find ##I_n##

b) Hence the obtain

$$lim_{n→∞}(I_{n-1}+I_n+I_{n+1}+...+I_{2n-2})=\int_0^W\frac{X}{Y+x}dx$$

Find X,Y,Z.

## Homework Equations

b) I think we may use Riemann sum, and the answer in a, help we in b, but I can't do it.

## The Attempt at a Solution

a) ##I_n=\int_{0}^{\pi/2}f_n(x)sin2xdx=2\int_{0}^{\pi/2}sin^{n+1}cosxdx=2\int_{0}^{\pi/2}sin^{n+1}d(sinx)##

$$I_n=\frac{2}{n+2}$$

b) Dont have any attempt.

I need some books about the calculus I that have some problems like this

Thank you very much

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