<s>[INST] Prove that $\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$ Prove that $$\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$$
My try


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*$n=0$ then $\frac{\sin x}{x}=\frac{1}{x} \int_0^x \cos y\,dy$ (true)

*Assuming $n=k$ (true)

*Prove $n=k+1$

 [/INST] Let $f_n(x)=\int_0^x y^n \cos(y+n\pi/2) dy$.  Then, using integration by parts,
$$\int_0^z \frac{1}{x^{n+1}} f_n(x) dx =
 -\frac{1}{nz^n} f_n(z)
 + \frac{1}{n} \sin(z+n\pi/2).$$
Now, again using integration by parts, this time on $f_n(z)$, it follows that the right hand side equals
$$\frac{1}{z^n} \int_0^z y^{n-1} \sin(y+n\pi/2)dy.$$
Note that $\sin(y+n\pi/2) = \cos(y+(n-1)\pi/2)$, which follows from $\sin(a+b)=\sin a \cos b + \sin b \cos a$.  
Hence,
$$\int_0^z \frac{1}{x^{n+1}} f_n(x) dx = \frac{1}{z^n} f_{n-1}(z).$$
Now use your induction trick.
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