How can I solve this integral, that is find f(x)? $$ f(x)=\int_{1}^{x} \left(\cos\left( \frac{\pi r}{t}\right)+\cos(\pi t)\right)^{N}dt $$ Could it be possible to obtain a closed form of $f(x)$? Alternatively, could it be possible to find an approximation of this integral using the Taylor series expansion of the integrand and integrate it piece-wise?
Having $$ g(t)=\left(\cos\left( \frac{\pi r}{t}\right)+cos(\pi t)\right)^{N} $$ the Taylor series is
$$ g(t) = \sum_{n=0}^{\infty} \frac{g^{(n)}(a)}{n!} (t-a)^n $$
Can I compute the derivatives value up to a certain degree $M$, and use the integral of nth-power in order to estimate this integral?
That is the indefinite integral should be something like: $$ \int g(t) dt = C+\sum_{n=0}^{\infty} \frac{g^{(n)}(a)}{n!} \frac{(t-a)^{n+1}}{n+1} $$
Does it converge? Can I go by this way? Others integration methods are error prones. Is it possible to evaluate it at arbitrary precision?