41.

The angle between the planes 3x + 4y + 5z = 3 and 4x - 3y + 5z = 9 is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

The vector equation of the plane through the point (2, 1, - 1) and parallel to the plane r - (i + 3j - k) = 0 is

• r . (i + 9j + 11k) = 6

• r . (i - 9j + 11k) = 4

• r . (i + 3j - k) = 6

• r . (i + 3j - k) = 4

If the foot of the perpendicular drawn from the point (5, 1, - 3) to a plane is (1, - 1, 3), then the equation of the plane is

• 2x + y - 3z + 8 = 0

• 2x + y + 3z + 8 = 0

• 2x - y - 3z + 8 = 0

• 2x - y + 3z + 8 = 0

The equation of the plane through the line of intersection of the planes x - y + z + 3 = 0 and x + y + 22 + 1 = 0 and parallel to x-axis is

• 2y - z = 2

• 2y + z = 2

• 4y + z = 4

• y - 2z = 3

$\int \frac{5\mathrm{x} \mathrm{dx}}{{\left(1 - \mathrm{x}\right)}^3}$ is equal to

• $\frac{5}{2{\left(\mathrm{x} - 1\right)}^2} - \frac{5}{\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{2{\left(\mathrm{x} - 1\right)}^2} + \frac{5}{\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{3{\left(\mathrm{x} - 1\right)}^2} + \frac{5}{2\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{3{\left(\mathrm{x} - 1\right)}^2} - \frac{5}{2\left(\mathrm{x} - 1\right)} + \mathrm{C}$

$\int \frac{\mathrm{dx}}{\mathrm{x} - \sqrt{\mathrm{x}}}$ is equal to

• $2\log \left|\sqrt{\mathrm{x}} - 1\right| + \mathrm{C}$

• $2\log \left|\sqrt{\mathrm{x}} + 1\right| + \mathrm{C}$

• $\log \left|\sqrt{\mathrm{x}} - 1\right| + \mathrm{C}$

• $\frac{1}{2}\log \left|\sqrt{\mathrm{x}} + 1\right| + \mathrm{C}$

$\int \frac{\mathrm{dx}}{4{\sin }^2\left(\mathrm{x}\right) + 3{\cos }^2\left(\mathrm{x}\right)}$

• $\frac{\sqrt{3}}{4}{\tan }^{-1}\left(\frac{2\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{3}}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{3}}{\tan }^{-1}\left(\frac{2\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

$\int \frac{\sec \left(\mathrm{x}\right)\mathrm{dx}}{\sqrt{\cos \left(2\mathrm{x}\right)}}$ is equal to

• 2${\sin }^{-1}\left(\tan \left(\mathrm{x}\right)\right)$

• ${\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{2}\right) + \mathrm{C}$

• ${\sin }^{-1}\left(\tan \left(\mathrm{x}\right)\right)$

• $\frac{\sqrt{3}}{2}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

$\int \frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}}\left(\mathrm{xlog}\left(\mathrm{x}\right) + 1\right)\mathrm{dx}$ is equal to

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{C}$

• ${\mathrm{xe}}^{\mathrm{x}}\log \left|\mathrm{x}\right| + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\log \left|\mathrm{x}\right| + \mathrm{C}$

• x(e^x + $\log \left|\mathrm{x}\right|$) + C

$\int \frac{1 + \log \left(\mathrm{x}\right)}{{\left(1 + \mathrm{x} \log \left(\mathrm{x}\right)\right)}^2}$dx is equal to

• $\frac{1}{1 + \mathrm{xlog}\left|\mathrm{x}\right|} + \mathrm{C}$

• $\frac{1}{1 + \log \left|\mathrm{x}\right|}$ + C

• $\frac{- 1}{1 + \mathrm{xlog}\left|\mathrm{x}\right|} + \mathrm{C}$

• $\log \left|\frac{1}{1 + \log \left|\mathrm{x}\right|}\right|$ + C