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Integrals

Multiple Choice Questions

621.

$\int_{- 1}^{1} ({ax}^{3} + bx) dx = 0$ for

any value of a and b
a > 0, b > 0 only
a > 0, b > 0 only
a < 0, b > 0 only

622.

Using the Trapezoidal rule, the approximate value of $\int_{1}^{4} ydx$

x 1 2 3 4

y 0.7111 0.7222 0.7333 0.7444

0.1833
1.1833
2.1833
3.1833

623.

$\int \frac{dx}{1 - \cos (x) - \sin (x)} is equal to$

$\log (|\begin{matrix} 1 + \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \tan (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 + \tan (\frac{x}{2}) \end{matrix}|) + c$

624.

$\int \frac{dx}{7 + 5 \cos (x)} is equal to$

$\frac{1}{\sqrt{3}} \tan^{- 1} (\frac{1}{\sqrt{3}} \tan \frac{x}{2}) + c$
$\frac{1}{\sqrt{6}} \tan^{- 1} (\frac{1}{\sqrt{6}} \tan \frac{x}{2}) + c$
$\frac{1}{7} \tan^{- 1} (\tan \frac{x}{2}) + c$
$\frac{1}{4} \tan^{- 1} (\tan \frac{x}{2}) + c$

625.

$\int \frac{3^{x} dx}{\sqrt{9^{x} - 1}} is equal to$

$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} - \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (9)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 9^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$

626.

$\int_{2}^{3} \frac{dx}{x^{2} - x is equal to}$

$\log (\frac{2}{3})$
$\log (\frac{4}{3})$
$\log (\frac{8}{3})$
$\log (\frac{1}{4})$

627.

$\int_{\frac{- π}{2}}^{\frac{π}{2}} \sin^{4} (x) \cos^{6} (x) dx is equal to$

$\frac{3 π}{128}$
$\frac{3 π}{256}$
$\frac{3 π}{572}$
$\frac{3 π}{64}$

628.

The approximate value of $\int_{2}^{9} x^{2} dx$ by using trapezoidal rule with 4 equal intervals, is

248
242.5
242.8
243

629.

$A minimum value of \int_{0}^{x} {te}^{t^{2}} dt is$

630.
$\int \frac{1 + x + \sqrt{x + x^{2}}}{\sqrt{x} + \sqrt{1 + x}} dx is equal to$

$\frac{1}{2} \sqrt{1 + x} + C$

$\frac{2}{3} {(1 + x)}^{\frac{3}{2}} + C$

$\sqrt{1 + x} + C$

$2 {(1 + x)}^{\frac{3}{2}} + C$

$\frac{2}{3} {(1 + x)}^{\frac{3}{2}} + C$

$\int \frac{1 + x + \sqrt{x + x^{2}}}{\sqrt{x} + \sqrt{1 + x}} dx = \int \frac{(1 + x) + \sqrt{x (1 + x)}}{\sqrt{x} + \sqrt{1 + x}} dx = \frac{\sqrt{1 + x} (\sqrt{1 + x} + \sqrt{x})}{\sqrt{x} + \sqrt{1 + x}} dx = \int \sqrt{1 + x} dx = \frac{{(1 + x)}^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} {(1 + x)}^{\frac{3}{2}} + C$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

630. ∫1 + x + x + x2x + 1 + xdx is equal to 121 + x + C 231 + x32 + C 1 + x + C 21 + x32 +C

630.
$\int \frac{1 + x + \sqrt{x + x^{2}}}{\sqrt{x} + \sqrt{1 + x}} dx is equal to$

$\frac{1}{2} \sqrt{1 + x} + C$

$\frac{2}{3} {(1 + x)}^{\frac{3}{2}} + C$

$\sqrt{1 + x} + C$

$2 {(1 + x)}^{\frac{3}{2}} + C$