Evaluate: ${\int }_2^3 \frac{1}{\mathrm{x}} \mathrm{dx}$

Evaluate: $\int$ sin x sin 2x sin 3x dx

Evaluate: $\int \frac{2}{( 1 - \mathrm{x} ) ( 1 + {\mathrm{x}}^2 )} \mathrm{dx}$

Prove that  ${\int }_0^{\frac{\mathrm{\pi }}{4}} \sqrt{ \tan \mathrm{x}} + \sqrt{ \cot \mathrm{x}} \mathrm{dx} = \sqrt{2}. \frac{\mathrm{\pi }}{2}$

Evaluate  ${\int }_1^3 2 {\mathrm{x}}^2 + 5 \mathrm{x} \mathrm{dx}$   as a limit of sum.

The integral  is equal to 

287.

Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (−1, −2), then which one of the following is a vertex of this rhombus?

• (−3, −9)

• (−3, −8)

• (1/3, -8/3)

• (1/3, -8/3)

The centres of those circles which touch the circle, x²+y²−8x−8y−4=0, externally and also touch the x-axis, lie on:

• a circle

• an ellipse which is not a circle

• a hyperbola.

• a hyperbola.

The integral 

The integral   is equal to 

• 2

• 4

• 1

• 1