# Solving $\int\sqrt{25\sin^2(5x)+49\cos^2(7x)}\,dx$

I have tried solving the following integral using Calc I & II methods and couldn't find an answer. I would like to know the techniques for solving such an integral.

$$\int\sqrt{25\sin^2(5x)+49\cos^2(7x)}dx$$

• If it does Wolfram doesn't know about it. Commented Nov 2, 2017 at 15:12
• It is the arc length of a Lissajous curve and it is very likely related to (hyper-)elliptic integrals. What is the original purpose of computing such primitive? Commented Nov 2, 2017 at 21:20
• Most likely there is no closed formula, as pointed out in problem 10 in here:math.harvard.edu/~knill/teaching/summer2011/handouts/… Commented Nov 3, 2017 at 9:49
• wolframalpha.com/… Commented Sep 9 at 16:03

$$\int\sqrt{25\sin^2(5x)+49\cos^2(7x)}\,dx$$ resembles the integral for the arc length of: $$\overrightarrow{r}(t)=\left\langle\cos(5t),\sin(7t) \right\rangle$$
Here $$\overrightarrow{r}(t)$$ is a Lissajous curve, which is a curve defined as the parametric equations: $$x=A\sin(at+\delta),\;\;y=B\sin(bt)$$ where $$t$$ is the parameter and $$A, a, B, b,\delta$$ are constants. We can derive the length of the curve fairly easily: $$L=\int\sqrt{(dx)^2+(dy)^2}=\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt=$$ $$\sqrt{\int(Aa\cos(at+\delta))^2+(Bb\cos(bt))^2}\,dt$$