<s>[INST] Name of the rule allowing the exchanging $\sin$ and $\cos$ in integrals with limits $0$ and $\pi/2$? As in $0$ to $\frac{\pi}{2}$ limits the area under curve of $\sin \theta$ and $\cos \theta$ are same, so in integration if the limits are from $0$ to $\frac{\pi}{2}$ we can replace $\sin \theta$ with $\cos \theta$ and vice versa. Example-
\begin{align*}
   \int\limits_{0}^{\frac{\pi}{2}} \frac{\sin^3x-\cos x}{\cos^3x-\sin x} dx &=\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin^3x-\sin x}{\sin^3x-\sin x} dx\\
&=\int\limits_{0}^{\frac{\pi}{2}}dx\\
&=\frac{\pi}{2}
\end{align*}
I want to know what the name of this rule.
 [/INST] This is not a specific rule. It is the property of definite integral: $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$ i.e. substitute $x=a+b-x$ everywhere in the integrand as follows
$$\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin^3x-\cos x}{\cos^3x-\sin x} dx$$
$$=\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin^3\left(\frac{\pi}{2}-x\right)-\cos\left(\frac{\pi}{2}-x\right)}{\cos^3\left(\frac{\pi}{2}-x\right)-\sin \left(\frac{\pi}{2}-x\right)} dx$$
$$=\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos^3x-\sin x}{\sin^3x-\cos x} dx$$
Using above property in this case is not useful because it's an improper integral.
</s>