If \(\int \cos ^{\frac{3}{5}} x \cdot \sin ^3 x \mathrm{~d} x=\frac{-1}{\mathrm{~m}} \cos ^{\mathrm{m}} x+\frac{1}{\mathrm{n}} \cos ^{\mathrm{n}} x+\mathrm{c}\), (where \(c\) is the constant of integration), then \((\mathrm{m}, \mathrm{n})=\)

Let
\(\mathbf{I} =\int \cos ^{\frac{3}{5}} x \sin ^3 x \mathrm{~d} x \)
\( =\int \cos ^{\frac{3}{5}} x\left(1-\cos ^2 x\right) \sin x \mathrm{~d} x \)
\( =\int \cos ^{\frac{3}{5}} x \sin x \mathrm{~d} x-\int \cos ^{\frac{13}{5}} x \sin x \mathrm{~d} x\)
Let \(\cos x=\mathrm{t} \Rightarrow-\sin x \mathrm{~d} x=\mathrm{dt}\)
\(\therefore I =-\int t^{\frac{3}{5}} d t+\int t^{\frac{13}{5}} d t \)
\( =\frac{-1}{\left(\frac{8}{5}\right)} t^{\frac{8}{5}}+\frac{1}{\left(\frac{18}{5}\right)} t^{\frac{13}{5}}+c \)
\( =\frac{-1}{\left(\frac{8}{5}\right)} \cos ^{\frac{8}{5}} x+\frac{1}{\left(\frac{18}{5}\right)} \cos ^{\frac{13}{5}} x+\mathrm{c}\)
Comparing with \(\frac{-1}{\mathrm{~m}} \cos ^{\mathrm{m}} x+\frac{1}{\mathrm{n}} \cos ^{\mathrm{n}} x+\mathrm{c}\), we get \(\mathrm{m}=\frac{8}{5}, \mathrm{n}=\frac{18}{5}\)