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Integrals

Multiple Choice Questions

21.

What is the solution of the differential equation

$\log (\frac{dy}{dx}) - a = 0$

y = xe^a + c
x = ye^a + c
$y = \ln (x) + c$
$x = \ln (y) + c$

22.

$\int_{e^{- 1}}^{e^{- 2}} |\frac{\log (x)}{x}| d x = ?$

$\frac{3}{2}$
$\frac{5}{2}$
3
4

23.

What is $\int \tan^{- 1} (\sec (x) + \tan (x)) dx = ?$

$\frac{πx}{4} + \frac{x^{2}}{4} + c$
$\frac{πx}{2} + \frac{x^{2}}{4} + c$
$\frac{πx}{4} + \frac{{πx}^{2}}{4} + c$
$\frac{πx}{4} - \frac{x^{2}}{4} + c$

24.

What is $\int_{0}^{2 π} \sqrt{1 + \sin (\frac{x}{2})} d x = ?$

25.

$\int (\ln {(x)}^{- 1}) dx - \int (\ln {(x)}^{- 2}) dx = ?$

$x {(\ln {(x)}^{})}^{- 1} + c$
$x {(\ln (x))}^{- 2}$ = c
$xln (x) + c$
$x {(\ln (x))}^{2} + c$

26.
The value of $\int_{0}^{\frac{π}{4}} \sqrt{\tan (x)} d x + \int_{0}^{\frac{π}{4}} \sqrt{\cot (x)} d x = ?$

$\frac{π}{4}$

$\frac{π}{2}$

$\frac{π}{2 \sqrt{2}}$

$\frac{π}{\sqrt{2}}$

$\frac{π}{\sqrt{2}}$

$\int_{0}^{\frac{π}{4}} \sqrt{\tan (x)} d x + \int_{0}^{\frac{π}{4}} \sqrt{\cot (x)} d x = \int_{0}^{\frac{π}{4}} (\sqrt{\tan (x)} + \sqrt{cott (x)}) d x = \int_{0}^{\frac{π}{4}} \frac{\sin (x) + \cos (x)}{\sqrt{\sin (x) \cos (x)}} dx = \sqrt{2} \int_{0}^{\frac{π}{4}} \frac{(\sin (x) + \cos (x))}{\sqrt{1 - {(\sin (x) - \cos (x))}^{2}}} dx Putting \sin (x) - \cos (x) = t \Rightarrow dt = (\sin (x) + \cos (x)) dx when x = 0, t = - 1 and x = \frac{π}{4}, t = 0 = \sqrt{2} \int_{- 1}^{0} \frac{1}{\sqrt{1 - t^{2}}} d t = \sqrt{2} \int_{- 1}^{0} {[\sin^{- 1} (t)]}_{- 1}^{0} = \sqrt{2} [0 - (- π / 2)] = \frac{π}{\sqrt{2}}$