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

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

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

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

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

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

$Given that \int (e^{\log (x)} + \sin (x)) \cos (x) dx = \int (x + \sin (x)) \cos (x) dx = \int (xcos (x) + \sin (x) \cos (x)) dx = \int (xcos (x)) dx + \int (\sin (x) \cos (x)) dx = (xsin (x) + \cos (x) . 1) + \int (\frac{2}{2} \sin (x) \cos (x)) dx + c = xsin (x) + \cos (x) + \int (\frac{\sin (2 x)}{2}) dx + c = xsin (x) + \cos (x) + \frac{1}{2} (\frac{- \cos (2 x)}{2}) + c = xsin (x) + \cos (x) - \frac{\cos (2 x)}{4} + c = xsin (x) + \cos (x) - \frac{1 - 2 \sin^{2} (x)}{4} + c = xsin (x) + \cos (x) - \frac{1}{4} + \frac{2 \sin^{2} (x)}{4} + c = xsin (x) + \cos (x) + \frac{\sin^{2} (x)}{2} + (c - \frac{1}{4}) = xsin (x) + \cos (x) + \frac{\sin^{2} (x)}{2} + C$