Find $$\int_{0}^{\frac{\pi}{2}} sin^5\theta \ \ d\theta$$ Hence, with reference to graphs of circular functions, find $$\int_{0}^{\frac{\pi}{2}} cos^5\theta \ \ d\theta$$ ... explaining your reasoning.
The first integral is not too difficult to compute and I was successfully able to get $\frac{8}{15}$. However, the issue is with the next part. This is a non-calculator question, and, as becomes obvious by graphing the two, the second integral is identical in value due to the symmetry of the two graphs.
As someone who has no idea what either of ${sin}^5\theta$ or ${cos}^5\theta$ look like, how could I answer the second part without having to compute it manually?
Note: The time given for the second part of the question (as this was a two part questions instead of one, as opposed to how I've shown it) was less than that of the first part, implying that the knowledge of such symmetry of graph was expected, and that was what was being tested.
Using the identity $\cos \theta = \sin(\pi/2 - \theta)$ lets you use $u$-substitution to relate the second integral to $$\int_0^{\pi/2} \sin^5 u\,du = \int_0^{\pi/2} \sin^5 \theta\,d\theta, $$ which you already know.