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Integrals

Multiple Choice Questions

731.

$If \int (e^{2 x} + 2 e^{x} - e^{- x} - 1) e^{(e^{x} + e^{- x})} dx = g (x) e^{(e^{x} + e^{- x})} + C, where c is a constant of integration, then g (0) is$

732.

The value of $\int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{1}{1 + e^{\sin (x)}} d x is :$

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

733.

$If \int \frac{\cos (θ)}{5 + 7 \sin (θ) - 2 \cos^{2} (θ)} dθ = {Alog}_{e} |B (θ)| + C, where C is a constant of integration, then \frac{B (θ)}{A} can be :$

$\frac{2 \sin (θ) + 1}{5 (\sin (θ) + 3)}$
$\frac{5 (2 \sin (θ) + 1)}{\sin (θ) + 3}$
$\frac{2 \sin (θ) + 1}{\sin (θ) + 3}$
$\frac{5 (\sin (θ) + 3)}{2 \sin (θ) + 1}$

734.
The general solution of the differential equation
$\sqrt{1 + x^{2} + y^{2} + x^{2} y^{2}} + xy \frac{dy}{dx} = 0 (where C is constant of integration)$

$\sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2} + 1}}) + C$

$\sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2}} - 1}) + C$

$\sqrt{1 + y^{2}} - \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2} + 1}}) + C$

$\sqrt{1 + y^{2}} - \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2} - 1}}) + C$

$\sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log_{e} (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2}} - 1}) + C$

$\int \frac{y}{\sqrt{1 + y^{2}}} dy = \int - \frac{\sqrt{1 + x^{2}}}{x} dx \int \frac{2 y}{2 \sqrt{1 + y^{2}}} dy = \int - \frac{1 + x^{2}}{x \sqrt{1 + x^{2}}} dx \Rightarrow \sqrt{1 + y^{2}} = - \int \frac{x}{\sqrt{1 + x^{2}}} dx - \int \frac{x}{x^{2} \sqrt{1 + x^{2}}} dx \Rightarrow \sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = - \int \frac{t}{(t^{2} - 1) t} dt \Rightarrow \frac{1}{2 (1)} \log (\frac{t + 1}{t - 1}) + C \Rightarrow \sqrt{1 + y^{2}} + \sqrt{1 + x^{2}} = \frac{1}{2} \log (\frac{\sqrt{1 + x^{2}} + 1}{\sqrt{1 + x^{2}} - 1}) C$

735.

$If I_{1} = \int_{0}^{1} {(1 - x^{50})}^{100} d x and I_{2} = \int_{0}^{1} {(1 - x^{50})}^{101} d x I_{2} = {αI}_{1} Then α = ?$

$\frac{5049}{5050}$
$\frac{5051}{5050}$
$\frac{5050}{5051}$
$\frac{5050}{5049}$

736.

$The integral \int_{1}^{2} e^{x} . x^{2} (2 + \log_{e} (x)) d x = ?$

e(2e - 1)
e(4e + 1)
4e² - 1
e(4e - 1)

737.

The common difference of the A.P. b₁,b₂,....,bm is 2 more than common difference of A.P. a₁,a₂,...,a_n. If a₄₀= –159, a₁₀₀= – 399 and b₁₀₀= a₇₀, then b₁is equal to :

127
81
- 127
- 81