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Integrals

Multiple Choice Questions

41.

What is $\int \frac{dx}{x {(1 + \ln (x))}^{n}}$ equal to (n ≠ 1) ?

$\frac{1}{(n - 1) {(1 + \ln (x))}^{n - 1}} + c$
$\frac{1 - n}{{(1 + \ln (x))}^{^{1 - n}}} + c$
$\frac{n + 1}{{(1 + \ln (x))}^{^{n + 1}}} + c$
$\frac{1}{(n - 1) {(1 + \ln (x))}^{^{n - 1}}} + c$

42.

What is $\int \frac{dx}{2 x^{2} - 2 x + 1} = ?$

$\frac{\tan^{- 1} (2 x - 1)}{2} + c$
2 $\tan^{- 1} (2 x - 1) + c$
$\frac{\tan^{- 1} (2 x + 1)}{2} + c$
$\tan^{- 1} (2 x - 1) + c$

43.

What is the solution of the differential equation $\ln (\frac{dy}{dx}) = x$

y = e^x + c
y = e^{- x} + c
y = ln(x) + c
y = 2ln(x) + c

44.

What is the value of k such that integration of $\frac{3 x^{2} + 8 - 4 k}{x}$ with respect to x, may be a rational function ?

45.

What is $\int \frac{dx}{x (x^{n} 1)} = ?$

$\frac{1}{n} \ln (\frac{x^{n}}{x^{n} + 1}) + c$
$\ln (\frac{x^{n} + 1}{x^{n}}) + c$
$\ln (\frac{x^{n}}{x^{n} + 1}) + c$
$\frac{1}{n} \ln (\frac{x^{n} + 1}{x^{n}}) + c$

46.

What is $\int (e^{\log (x)} + \sin (x)) \cos (x) dx = ?$

$\sin (x) + xcos (x) + \frac{\sin^{2} (x)}{2} + C$
$\sin (x) - xcos (x) + \frac{\sin^{2} (x)}{2} + C$
$x \sin (x) + \cos (x) + \frac{\sin^{2} (x)}{2} + C$
$x \sin (x) - \cos (x) + \frac{\sin^{2} (x)}{2} + C$

47.
What is the value of $\int_{0}^{\frac{π}{4}} (\tan^{3} (x) + \tan (x)) d x ?$

$\frac{1}{4}$ $\frac{1}{4}$

$\frac{1}{2}$

1

2

$\frac{1}{2}$

$\int_{0}^{\frac{π}{4}} (\tan^{3} (x) + \tan (x)) d x = \int_{0}^{\frac{π}{4}} \tan (x) (\tan^{2} (x) + 1) d x = \int_{0}^{\frac{π}{4}} \tan (x) \sec^{2} (x) dx = \int_{0}^{\frac{π}{4}} \tan (x) \sec (x) . \sec (x) dx Assuming \sec (x) = t So \tan (x) . \sec (x) dx = dt Putting above = \int tdt = \frac{t^{2}}{2} Putting back \sec (x) = t = {[\frac{\sec^{2} (x)}{2}]}_{0}^{\frac{π}{4}} = \frac{1}{2} [{(\sqrt{2})}^{2} - 1] = \frac{1}{2} (1) = \frac{1}{2}$