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Integrals

Multiple Choice Questions

701.

$\int \frac{x^{2} dx}{{(xsin (x) + \cos (x))}^{2}}$ is equal to

$\frac{\sin (x) + \cos (x)}{xsin (x) + \cos (x)} + c$
$\frac{x \sin (x) - \cos (x)}{xsin (x) + \cos (x)} + c$
$\frac{\sin (x) - xcos (x)}{xsin (x) + \cos (x)} + c$
None of these

702.

If f(x) = $Asin (\frac{πx}{2}) + B, f' (\frac{1}{2}) = \sqrt{2}$ and $\int_{0}^{1} f (x) dx = \frac{2 A}{π}$ , then A and B are

$\frac{π}{2}, \frac{π}{2}$
$\frac{2}{π}, \frac{3}{π}$
$0, - 4 π$
$\frac{4}{π}, 0$

703.

Let g(x) = $\int_{0}^{x} f (t) dt$ , where f is such that $\frac{1}{2} \leq f (x) \leq 1$ for t $\in$ [0, 1] and $0 \leq f (t) \leq \frac{1}{2}$ for t $\in$ [1, 2]. Then, g(2) satisfies the inequality

$- \frac{3}{2} \leq g (2) < \frac{1}{2}$
$0 \leq g (2) < 2$
$\frac{1}{2} \leq g (2) < \frac{3}{2}$
2 < g(2) < 4

704.

$\int \frac{dx}{\sin (x) - \cos (x) + \sqrt{2}}$ is equal to

$- \frac{1}{\sqrt{2}} \tan (\frac{x}{2} + \frac{π}{8}) + C$
$\frac{1}{\sqrt{2}} \tan (\frac{x}{2} + \frac{π}{8}) + C$
$\frac{1}{\sqrt{2}} \cot (\frac{x}{2} + \frac{π}{8}) + C$
$\frac{1}{\sqrt{2}} \cot (\frac{x}{2} + \frac{π}{8}) + C$

705.

$\int e^{\frac{x}{2}} \sin (\frac{x}{2} + \frac{π}{4}) dx$ is equal to

$e^{\frac{x}{2}} \cos (\frac{x}{2}) + C$
$\sqrt{2} e^{\frac{x}{2}} \cos (\frac{x}{2}) + C$
$e^{\frac{x}{2}} \sin (\frac{x}{2}) + C$
$\sqrt{2} e^{\frac{x}{2}} \sin (\frac{x}{2}) + C$

706.
The solution of the differential equation xdy - ydx = $(\sqrt{x^{2} + y^{2}}) dx$ is

$y - \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

$y + \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

$y + \sqrt{x^{2} + y^{2}} + {Cx}^{2} = 0$

None of the above

$y + \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

$xdy - ydx = \sqrt{x^{2} + y^{2}} dx xdy = (y + \sqrt{x^{2} + y^{2}}) dx \frac{dy}{dx} = \frac{y + \sqrt{x^{2} + y^{2}}}{x} It is homogeneous equation Put y = vx \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx} v + x \frac{dv}{dx} = \frac{vx + x \sqrt{1 + v^{2}}}{x} = \frac{v + \sqrt{1 + v^{2}}}{1} \Rightarrow x \frac{dv}{dx} = \sqrt{1 + v^{2}} \Rightarrow \int \frac{dv}{\sqrt{1 + v^{2}}} = \int \frac{dx}{x} \Rightarrow \log (v + \sqrt{1 + v^{2}}) = \log (x) + \log (C) \Rightarrow \log (\frac{y}{x} + \frac{\sqrt{x^{2} + y^{2}}}{x}) = \log (x) + \log (C) \Rightarrow (y + \sqrt{x^{2} + y^{2}}) - \log (x) = \log (x) + \log (C) \Rightarrow \log (y + \sqrt{x^{2} + y^{2}}) = \log (x^{2} C) ∴ y + \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

707.

If $\int_{0}^{t^{2}} x f (x) dx = \frac{2}{5} t^{5}, t > 0, then f (\frac{4}{25})$ is

$\frac{2}{5}$
$\frac{5}{2}$
$- \frac{2}{5}$
None of these

708.

The value of interal I = $\int \frac{\sin (x) + \cos (x)}{\sqrt{1 + \sin (2 x)}} dx$ is

$\sqrt{1 + \cos (2 x)}$
$\sqrt{x}$
x
$\sqrt{1 + 2 x}$

709.

The value of interal $\int_{- \frac{3}{2}}^{\frac{3}{2}} \sin^{3} (x) \cos^{3} (x) dx$ is

0
1/2
1
None of these

710.

$\int \frac{x^{e - 1} + e^{x - 1}}{x^{e} + e^{x}} dx$ is equal to

$\frac{1}{e} \log (x^{e} - e^{x}) + c$
$\frac{1}{e} \log (x^{e} + e^{x}) + c$
$\frac{1}{e} \log (e^{x} - x^{e}) + c$
None of the above

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Multiple Choice Questions

706. The solution of the differential equation xdy - ydx = x2 + y2dx is y - x2 + y2 = Cx2 y + x2 + y2 = Cx2 y + x2 + y2 + Cx2 = 0 None of the above

706.
The solution of the differential equation xdy - ydx = $(\sqrt{x^{2} + y^{2}}) dx$ is

$y - \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

$y + \sqrt{x^{2} + y^{2}} = {Cx}^{2}$

$y + \sqrt{x^{2} + y^{2}} + {Cx}^{2} = 0$

None of the above