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Integrals

Multiple Choice Questions

651.

$\int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan^{3} (x)} = ?$

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

652.

$\int_{- 1}^{1} \frac{\cos (hx)}{1 + e^{2 x}} d x = ?$

0
1
$\frac{e^{2} - 1}{2 e}$
$\frac{e^{2} + 2}{2 e}$

653.

$If \int \frac{e^{x} - 1}{e^{x} + 1} dx = f (x) + c, then f (x) is equal to$

$2 \log (e^{x} - 1)$
$2 \log (e^{2 x} - 1)$
$2 \log (e^{x} + 1) - x$
$2 \log (e^{2 x} + 1)$

654.

$\int \tan^{- 1} (\sqrt{\frac{1 - x}{1 + x}}) dx = ?$

$\frac{1}{2} ({xcos}^{- 1} (x) - \sqrt{1 - x^{2}}) + c$
$\frac{1}{2} ({xcos}^{- 1} (x) + \sqrt{1 - x^{2}}) + c$
$\frac{1}{2} (x \sin^{- 1} (x) - \sqrt{1 - x^{2}}) + c$
$\frac{1}{2} (x \sin^{- 1} (x) + \sqrt{1 - x^{2}}) + c$

655.

$\frac{\sin (x) + 8 \cos (x)}{4 \sin (x) + 6 \cos (x)} dx = ?$

$x + \frac{1}{2} \log (4 \sin (x) + 6 \cos (x)) + c$
$2 x + \log (2 \sin (x) + 3 \cos (x)) + c$
$x + \log (2 \sin (x) + 3 \cos (x)) + c$
$\frac{1}{2} \log (4 \sin (x) + 6 \cos (x)) + c$

656.

$If f (t) = \int_{- t}^{t} \frac{e^{- |x|}}{2} dx, then \lim_{t \to \infty} f (t) = ?$

1
$\frac{1}{2}$
0
- 1

657.
$\int_{0}^{2 π} \sin^{6} (x) \cos^{5} (x) dx = ?$

$2 π$

$\frac{π}{2}$

0

$- π$

$Let I = \int_{0}^{2 π} \sin^{6} (x) \cos^{5} (x) dx I = 2 \int_{0}^{π} \sin^{6} (x) \cos^{5} (x) dx [∵ f (2 π - x) = f (x)] Let f (x) = \sin^{6} (x) \cos^{5} (x) [∵ f (π - x) = - f (x)] f (π - x) = \sin^{6} (π - x) \cos^{5} (π - x) = \sin^{6} (x) (- \cos^{5} (x)) = - \sin^{6} (x) (\cos^{5} (x)) = - f (x) I = 0$