Answer: \(y=\sin5x\), so \(a=1\), \(b=5\), and \(\displaystyle \int_0^{\pi/5}y\,dx=\frac25\).

Answer: \(y=\sin5x\), so \(a=1\), \(b=5\), and \(\displaystyle \int_0^{\pi/5}y\,dx=\frac25\).

Use the identity

\(\operatorname{sec}^{2} 5x-\tan^{2} 5x=1.\)

Therefore

\(y=\frac{1}{\operatorname{cosec}5x}.\)

Since \(\operatorname{cosec}5x=\frac{1}{\sin5x}\),

\(y=\sin5x.\)

Hence \(a=1\) and \(b=5\).

Now

Integrating,

\(\int\sin5x\,dx=-\frac15\cos5x.\)

So

This gives