Question

Find : $\large\textstyle\int$$\frac{x^{4}+1}{x\left(x^{2}+1\right)^{2}}d x$.

Find : $\large\textstyle\int$$\large\frac{x^{2}\,d x}{x^{4}+x^{2}-12}\;.$

Using properties of determinants, prove that : $\begin{vmatrix}b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y\end{vmatrix}$ = 2 $\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}$ .

Using properties of determinants, prove that $\begin{vmatrix}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{vmatrix}$ = x³

Find the equation of tangent to the curve $y={\frac{x-7}{x^{2}-5x+6}}$ at the point, where it cuts the x-axis.

Three schools A, B and C organised a fete (mela) collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being ₹ 25, ₹ 100 and ₹ 50 respectively. The following table shows the number of articles of each type sold :

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\cot x=2x+x^{2}\cot x,$ x ≠ 0. Given that y = 0, when $x={\frac{\pi}{2}}\ .$