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Integrals

Multiple Choice Questions

681.

$If a > 0, then \int_{- π}^{π} \frac{\sin^{2} (x)}{1 + a^{x}} dx = ?$

$\frac{π}{2}$
$π$
$\frac{2 π}{2}$
a $π$

682.

$The value of the integral \int_{0}^{4} \frac{dx}{1 + x^{2}} obtained by using trapezoidal rule with h = 1 is$

$\frac{63}{85}$
$\tan^{- 1} (4)$
$\frac{108}{85}$
$\frac{113}{85}$

683.
$Let A = |\begin{matrix} 2 & e^{iπ} \\ - 1 & i^{2012} \end{matrix}|, C = \frac{d}{dx} (\frac{1}{x}) x = 1, D = \int_{e^{2}}^{1} \frac{dx}{x} . If the sum of two roots of the equation {Ax}^{3} + {Bx}^{2} + Cx - D = 0 is equal to zero, then B is equal to$

- 1

0

1

2

$Given, A = |\begin{matrix} 2 & e^{iπ} \\ - 1 & i^{2012} \end{matrix}| = |\begin{matrix} 2 & \cos (π) + isin (π) \\ - 1 & {(1^{4})}^{503} \end{matrix}| = |\begin{matrix} 2 & - 1 \\ - 1 & 1 \end{matrix}| = 2 - 1 = 1 C = \frac{d}{dx} {(\frac{1}{x})}_{x = 1} = {(- \frac{1}{x^{2}})}_{x = 1} = - 1 and D = \int_{e^{2}}^{1} \frac{dx}{x} = {[\log (x)]}_{e^{2}}^{1} = \log (1) - \log (e^{2}) = 0 - 2 . . . (i) Let α, β and γ are the roots of the equation {Ax}^{3} + {Bx}^{2} + Cx + D = 0 ∴ x^{3} + {Bx}^{2} - x + 2 = 0 . . . (ii) ∴ α + β + γ = - B \Rightarrow γ = - B (∵ α + β = 0 given) ∴ - B satisfies eq . (ii), ∴ {(- B)}^{3} + B^{2} + B + 2 = 0 \Rightarrow B = 2$

684.

$\int e^{x} (\frac{2 + \sin (2 x)}{1 + \cos (2 x)}) dx = ?$

$e^{x} \cot (x) + C$
$2 e^{x} \sec^{2} (x) + C$
$e^{x} \cos (2 x) + C$
$e^{x} \tan (x) + C$

685.

$If \int \frac{x - \sin (x)}{1 + \cos (x)} dx = xtan (\frac{x}{2}) + plog |\sec (\frac{x}{2})| + C, then p = ?$

686.

$If \int \frac{dx}{xlog (x - 2) \log (x - 3)} = I + C, then I = ?$

$\frac{1}{x} \log |\frac{\log (x - 3)}{\log (x - 2)}|$
$\log |\frac{\log (x - 3)}{\log (x - 2)}|$
$\log |\frac{\log (x - 2)}{\log (x - 3)}|$
$\log |\log (x - 3) \log (x - 2)|$

687.

$If \int_{0}^{b} \frac{dx}{1 + x^{2}} = \int_{b}^{\infty} \frac{dx}{1 + x^{2}}, then b = ?$

$\tan^{- 1} (\frac{1}{3})$
$\frac{\sqrt{3}}{2}$
$\sqrt{2}$
1

688.

The approximate value of $\int_{1}^{3} \frac{dx}{2 + 3 x}$ using Simpson's rule and dividing the interval [1, 3] into two equal parts is

$\frac{1}{3} \log (\frac{11}{5})$
$\frac{107}{110}$
$\frac{22}{110}$
$\frac{119}{440}$

689.

$If \int \frac{dx}{\sqrt{\sin^{3} (x) \cos (x)}} = g (x) + c, then g (x) = ?$

$- \frac{2}{\sqrt{\cot (x)}}$
$- \frac{2}{\sqrt{\tan (x)}}$
$\frac{2}{\sqrt{\cot (x)}}$
$\frac{2}{\sqrt{\tan (x)}}$

690.

$If \int \frac{dx}{(1 + \sqrt{x}) \sqrt{x - x^{2}}} = \frac{A \sqrt{x}}{\sqrt{1 - x}} + \frac{B}{\sqrt{1 - x}} + C, where C is real constant, then A + B = ?$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

683. Let A = 2eiπ- 1i2012, C= ddx1xx = 1,D = ∫e21dxx. If the sum of two roots of the equation Ax3 + Bx2 + Cx - D = 0 is equal to zero, then B is equal to - 1 0 1 2

683.
$Let A = |\begin{matrix} 2 & e^{iπ} \\ - 1 & i^{2012} \end{matrix}|, C = \frac{d}{dx} (\frac{1}{x}) x = 1, D = \int_{e^{2}}^{1} \frac{dx}{x} . If the sum of two roots of the equation {Ax}^{3} + {Bx}^{2} + Cx - D = 0 is equal to zero, then B is equal to$

- 1

0

1

2