Question

Let $f\left(x\right)=\left|\begin{array}{ccc}1& {\cos }^{2}x& {\cos }^{2}x\\ {\cos }^{2}x& {\cos }^{2}x& {\cot }^{2}x\\ \sec x& \cos x& {\sec }^{2}x+\cos x.{\text{cosec}}^{2}x\end{array}\right|.$ If $\underset{0}{\overset{\pi /2}{\int }}f\left(x\right)dx=m\pi +n$, then the value of $4m+15n$ is

Answer 1

$f\left(x\right)=\left|\begin{array}{ccc}1& {\cos }^{2}x& {\cos }^{2}x\\ {\cos }^{2}x& {\cos }^{2}x& {\cot }^{2}x\\ \sec x& \cos x& {\sec }^{2}x+\cos x.{\text{cosec}}^{2}x\end{array}\right|$

$R_3\to R_3-\sec x{R}_1$
$=\left|\begin{array}{ccc}1& {\cos }^{2}x& {\cos }^{2}x\\ {\cos }^{2}x& {\cos }^{2}x& {\cot }^{2}x\\ 0& 0& \frac{1}{{\cos }^{2}x}+\frac{\cos x}{{\sin }^{2}x}-\cos x\end{array}\right|$
$=[\frac{1}{{\cos }^{2}x}+\frac{\cos x}{{\sin }^{2}x}(1-{\sin }^{2}x\left)\right]({\cos }^{2}x-{\cos }^{4}x)$
$=[\frac{1}{{\cos }^{2}x}+\frac{{\cos }^{3}x}{{\sin }^{2}x}]({\cos }^{2}x-{\cos }^{4}x)$
$={\sin }^{2}x+{\cos }^{5}x$

$\underset{0}{\overset{\pi /2}{\int }}f\left(x\right)dx=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{2}x+{\cos }^{5}xdx$
$=\frac{\pi }{4}+\frac{8}{15}$

$\Rightarrow m=\frac{1}{4},n=\frac{8}{15}\Rightarrow 4m+15n=9$