Question

Find : $\textstyle\int{\frac{d x}{x^{2}+4x+8}}$

If x changes from 4 to 4·01, then find the approximate change in log x.

If A = $\left(\begin{array}{l l }3 & 1 \\ -1 & 2\end{array}\right)$ and I = $\left(\begin{array}{l l }1 & 0 \\ 0 & 1\end{array}\right)$ find k so that A² = 5A + kI.

For the curve y = 5x – 2x³, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3.

The position vectors of points A, B and C are $λ\stackrel{\wedge}{i}+3\stackrel{\wedge}{j}$,$12\stackrel{\wedge}{i}+μ\stackrel{\wedge}{j}$ and $11\stackrel{\wedge}{i}-3\stackrel{\wedge}{j}$ respectively. If C divides the line segment joining A and B in the ratio 3 : 1, find the values of λ and μ.

Find the integrating factor of the differential equation ${\frac{d y}{d x}}+y\ =\ {\frac{1+y}{x}}$

Find : $\textstyle\int e^{x}\,{\frac{\sqrt{1+\sin2x}}{1+\cos2\,x}}\,d x$.