Is it possible to do this integral $$ \int \sqrt{(\cos t - t \sin t)^2 + (1 - 5 \cos t)^2} dt$$ by hand using usual methods? I tried in Wolfram Alpha but it's showing "SlowLarge + constant".
Edit : I was asked to find the arc length of the portion of the curve given by the parametric equation : $$ x = t \cos t, \\ y = t - 5\sin t$$ and there was a graph attached to the question where I could see that the portion was lying between $x= -π$ and $x = π$.
In the question it was mentioned that "set up an integral. Then use a calculator to fing the length correct upto 4 decimals". But I was confused that whether it was telling me to integrate it entirely using calculator or just to use the calculator while putting values of the limits of $t$.
I set up the integral as $$ \int_{-π}^{π} \sqrt{(\cos t - t \sin t)^2 + (1 - 5 \cos t)^2} dt$$.
But I later realised that I mistakenly put the limits of $x$ in place of $t$. Someone in the comment also said that the limits of $t$ is also $-π$ to $π$. How can I evaluate this(the limits of $t$ from the limits of $x$ I mean), can anyone please help me?
Edit 2 :
The original question :
Very often, finding indefinite integral then applying the fundamental theorem of calculus is very different from calculating directly the definite integrals. In a computer, the latter is done by numerical integration while the formal requires symbolic calculation.
The integral you set up is correct since you calculated the derivatives and applied the length formula correctly.
If you use WolframAlpha, you can find: