Question

Solve the differential equation $(\tan^{-1}x-y)d x=(1+x^{2})\,d y$

Evaluate : $\large\textstyle\int{\frac{3x+1}{(x^{3}-x^{2}-x+1)}}d x\,.$

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

Find the points on the curve y = x³ at which the slope of the tangent is equal to y-coordinate of the point.

If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

Find the particular solution of the following differential equation : $(x\ +1){\frac{d y}{d x}}=2e^{-y}-1;$ y = 0 when x = 0.

Without expanding the determinant at any stage; prove that : $\begin{vmatrix}0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0\end{vmatrix}$ = 0.