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

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

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

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

$\int \frac{dx}{2 x^{2} - 2 x + 1} = \frac{1}{2} \int \frac{dx}{x^{2} - x + \frac{1}{2}} = 2 \int \frac{dx}{(x^{2} - x + \frac{1}{4} - \frac{1}{4} + \frac{1}{2})} = 2 \int \frac{dx}{{(x - \frac{1}{2})}^{2} + {(\frac{1}{2})}^{2}} = \frac{1}{2} \times \frac{1}{\frac{1}{2}} \tan^{- 1} (2 x - 1) = \tan^{- 1} (2 x - 1) + c$