Question

Evaluate: $\textstyle\int\cot{x}(cosec x-1)e^{x}d x$ .

Given A = $\left(\begin{array}{l l l }4 & 2 & 5 \\ 2 & 0 & 3 \\ -1 & 1 & 0\end{array}\right)$ , write the value of det. $(2A A^{-1}).$

If $A_{i j}$ is the cofactor of the element $a_{i j}$ of the determinant $\begin{vmatrix}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{vmatrix}$ , then write the value of $a_{32}A_{32}.$

Write the degree of the differential equation : $\left(\frac{d y}{d x}\right)^{4}+3x\frac{d^{2}y}{d x^{2}}=0.$

Write the value of : $\textstyle\int{\frac{\sec^{2}x}{\operatorname{cosec}^{2}x}}d x.$

Given $\textstyle\int e^{x}(\tan x+1)\sec x\,d x=e^{x}f(x)+c.$ Write f(x) satisfying the above.

Find : $\textstyle\int{\frac{\sin^{2}\!x-\cos^{2}\!x}{\sin^{2}\!x\cos^{2}\!x}}\,d x$