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

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

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

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

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

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

Let

$I = \int \frac{dx}{x (x^{n} + 1)} = \int \frac{x^{n - 1} dx}{x^{n} (x^{n} + 1)} Let x^{n} + 1 = t \Rightarrow x^{n - 1} dx = dt = \frac{1}{n} \int \frac{dt}{(t - 1) t} = \frac{1}{n} \int (\frac{1}{t - 1} - \frac{1}{t}) dt = \frac{1}{n} \ln (\frac{t - 1}{t}) + c = \frac{1}{n} \ln (\frac{x^{n}}{x^{n} + 1}) + c$