`f(x)=|(secx,cos x,sec^2x + cot x cosecx),(cos^2x,cos^2x,cosec^2 x),(1,cos^2x,cos^2x)|` then `int_0^(pi/2) f(x) dx=.....`

Answer

`int_0^(pi/2) f(x) dx=(frac{1}{3}-frac{pi}{4})`

Explanation

\begin{aligned}
&f(x)=\left|\begin{array}{ccc}
\sec x & \cos x & \sec ^{2} x+\cos x \cdot \operatorname{cosec}^{2} x \\
\cos ^{2} x & \cos ^{2} x & \operatorname{cosec} x \\
\sin ^{2} x & 0 & 0
\end{array}\right| \\
&\because C_{23}(-1) \\
&=\cos x-\sin ^{2} x-\cos ^{3} x \\
...