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Integrals

Multiple Choice Questions

321.

The integral $\int \sec^{\frac{2}{3}} (x) \csc^{\frac{4}{3}} (x) dx$ is equal to

$- 3 \cot^{\frac{- 1}{3}} (x) + C$
$- 3 \tan^{\frac{- 1}{3}} (x) + C$
$3 \tan^{\frac{- 1}{3}} (x) + C$
$- \frac{3}{4} \tan^{\frac{- 4}{3}} (x) + C$

322.

If $\int \frac{dx}{{(x^{2} - 2 x + 10)}^{2}} = A (\tan^{- 1} (\frac{x - 1}{3}) + \frac{f (x)}{x^{2} - 2 x + 10}) + C$ where C is a constant of integration, then

$A = \frac{1}{54}, f (x) = 3 (x - 1)$
$A = \frac{1}{54}, f (x) = 9 {(x - 1)}^{2}$
$A = \frac{1}{27}, f (x) = 9 (x - 1)$
$A = \frac{1}{81}, f (x) = 3 (x - 1)$

323.

The value of $\int_{0}^{2 π} [\sin (2 x) (1 + \cos (3 x))] dx$ , (where [t] denotes Greatest Integer Function)

$- 2 π$
$π$
$2 π$
$- π$

324.

If $\int x^{5} e^{- x^{2}} dx = g (x) e^{- 2} + c$ , where c is a constant of interation then g(- 1) is equal to :

- 1
1
$\frac{1}{2}$
$- \frac{5}{2}$

325.
The integral $\int_{\frac{π}{6}}^{\frac{π}{3}} \sec^{\frac{2}{3}} (\csc^{\frac{4}{3}} (x)) dx$ =

3^5/3 - 3^1/3

3^7/6 - 3^5/6

3^5/6 - 3^1/3

3^4/3 - 3^1/3

3^7/6 - 3^5/6

$\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sec^{2} (u)}{{(\tan (u))}^{\frac{4}{3}}} d x \int_{4 \sqrt{3}}^{\sqrt{3}} t^{- \frac{4}{3}} dt = - 3 {[t^{\frac{t}{3}}]}_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} = 3 [{(\sqrt{3})}^{\frac{1}{3}} - {(\sqrt{3})}^{\frac{1}{3}}] = 3^{\frac{7}{6}} - 3^{\frac{5}{6}}$