Integración

Question

Considerar: $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$

Buscar: $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.

Answer 1

Tenga en cuenta que $$ \int_{0}^{x}f(t)\ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^4}\int_{0}^{x}\sin(nt) \ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^4}\left( \frac{1}{n}-\frac{\cos(nx)}{n} \right)

$$\implies \int_{0}^{x}f(t) \ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^5}-\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^5}$$

$$\implies f(x)=\zeta(5)-\frac{1}{2}\sum_{n=1}^{\infty} \frac{e^{inx} }{n^5} - \frac{1}{2}\sum_{n=1}^{\infty} \frac{e^{-inx} }{n^5}$$

$$\implies f(x)=\zeta(5)-\frac{1}{2}( \operatorname{Li}_{5}(e^{ix})+ \operatorname{Li}_{5}(e^{-ix}) ),$$

donde $\operatorname{Li}_{s}(z)$ es la función del Polylogarithm.