Can the Taylor series be used to calculate this integral? The Wiki article here

says that non-elementary (definite) integrals can often be calculated using the Taylor series.  Can the Taylor series be used to calculate the integral of $$\ln\left[\left(\sin(x) + \frac{\sin(2x)}{2} +\frac {\sin(3x)}{3}\right)^2+0.1\right]$$ from $0$ to $2\pi$?  If so, how?
Sometimes a power series can be integrated term by term, i.e. sometimes $$\int_a^b\sum_{k=0}^\infty c_k(x-a)^k\ dx=\sum_{k=0}^\infty\int_a^b c_k(x-a)^k\ dx.$$ When is "sometimes"? When the series converges uniformly on $[a,b]$, for one.

Thus your first question reduces to whether the power series of the function converges uniformly on $[0,2\pi]$.
Sometimes a power series can be integrated term by term, i.e. sometimes $$\int_a^b\sum_{k=0}^\infty c_k(x-a)^k\ dx=\sum_{k=0}^\infty\int_a^b c_k(x-a)^k\ dx.$$ When is "sometimes"? When the series converges uniformly on $[a,b]$, for one.

Thus your first question reduces to whether the power series of the function converges uniformly on $[0,2\pi]$.