Integration by Parts

Q. $\int 2$ cos 2x In (tan⁡x )dx, is equal to

1. sin 2x In (tan x) – 2x + C
2. sin 2xIn (tan x) + 2x + C
3. sin xIn (tan x) – x + C
4. none of these

Q. $\int x\left\{f\right(x^2\left)g^{\prime \prime }\right(x^2)-f^{\prime \prime }(x^2\left)g\right(x^2\left)\right\}dx$

1. $f\left(x^2\right)g^{{}^{\prime }}\left(x^2\right)-g\left(x^2\right)f^{{}^{\prime }}\left(x^2\right)+c$
2. $\frac{1}{2}\left\{f\right(x^2\left)g\right(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$
3. $\frac{1}{2}\left\{f\right(x^2\left)g^{{}^{\prime }}\right(x^2)-g(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$
4. None of these

Q. If $\int \frac{se{c}^{2}x-2010}{si{n}^{2010}x}dx=\frac{P\left(x\right)}{si{n}^{2010}x}+C,$ then value of $P\left(\frac{\pi }{3}\right)$ is

1. $0$
2. $\frac{1}{\sqrt{3}}$
3. $\sqrt{3}$
4. $Noneofthese$

Q. $\int \frac{1-7co{s}^{2}x}{si{n}^{7}xco{s}^{2}x}dx=\frac{f\left(x\right)}{(sinx{)}^7}+C$, then f(x) is equal to

1. sin x
2. cos x
3. tan x
4. cot x

Q. Let $F\left(x\right)=\int e^{si{n}^{-1}x}(1-\frac{x}{\sqrt{1-x^2}})dx,andF\left(0\right)=1,ifF\left(\frac{1}{2}\right)=\frac{k{\sqrt{3e}}^{\frac{x}{6}}}{\pi }$, then k =

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{3}$

Q. If $\Phi \left(x\right)={lim}_{n\to \infty }\frac{x^n-x^{-n}}{x^n+x^{-n}},0<x<1,ϵN$, then $\int \left(si{n}^{-1}x\right)\left(\Phi \right(x\left)\right)dx$ is equal to

1. None of these
2. $xsi{n}^{-1}\left(x\right)+\sqrt{1-x^2}+C$
3. $-\left(xsi{n}^{-1}\right(x)+\sqrt{1-x^2})$
4. $xsi{n}^{-1x+\sqrt{1-x^2}+C}$

Q. If $\int si{n}^{-1}xco{s}^{-1}xdx=f^{-1}[\frac{\pi }{2}x-x{f}^{-1}(x)-2\sqrt{1-x^2}]+\frac{\pi }{2}\sqrt{1-x^2}+2x+C$ then f(x) is equal to

1. sin 3x
2. sin 4x
3. sin 2x
4. sin x

Q. If $\int f\left(x\right)dx=\Psi \left(x\right)$, then $\int x^{5}f\left(x^3\right)dx$ is equal to

1. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-\int x^2\Psi \left(x^3\right)dx+C$
2. $\frac{1}{3}\left[x^3\Psi \right(x^3)-\int x^2\Psi (x^3\left)dx\right]+C$
3. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-\int x^3\Psi \left(x^3\right)dx+C$
4. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-3\int x^3\Psi \left(x^3\right)dx+C$

Q. $\int e^{sinx}\left(\frac{xco{s}^{3}x-sinx}{co{s}^{2}x}\right)dx$, is equal to

1. $e^{sinx}(tanx+x)+C$
2. $e^{sinx}(x-secx)+C$
3. $e^{sinx}(secx+tanx)+C$
4. $Noneofthese$

Q. A car moves with a variable acceleration given by $a={\tan }^{-1}\left(t\right)$ where t is the time in seconds. Find the velocity of the car after 10 seconds, if it was initially at rest

1. $10{\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |{10}^2|$
2. $10{\tan }^{-1}(1+{10}^2)-\frac{1}{2}\ln |{10}^2|$
3. $10{\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |1+{10}^2|$
4. ${\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |1+{10}^2|$