CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 2 - MCQExams.com

$$ \int \sin ^{-1}(\cos x) d x  $$ is equal to
  • $$ \frac{\pi x}{2}+c $$
  • $$ \frac{\pi x^{2}}{2}+c $$
  • $$ \frac{\pi x-x^{2}}{2}+c $$
  • $$ \frac{\pi x+x^{2}}{2}+c $$
What is $$\displaystyle \int \dfrac{dx}{x(1 + ln x)^n}$$ equal to $$(n \neq 1)$$ ?
  • $$\dfrac{1}{(n - 1)(1 + ln x)^{n - 1}} + c$$
  • $$\dfrac{1 - n}{(1 + ln x)^{1- n}} + c$$
  • $$\dfrac{n + 1}{(1 + ln x)^{n+1}} + c$$
  • $$-\dfrac{1}{(n - 1)(1 + ln x)^{n-1}} + c$$
Evaluate $$\displaystyle \int xsecx.tanxdx=$$
  • $$x\sec x+\log|\tan(\pi/2+x/2)|+c$$
  • $$x\sec x-\log $$$$|\tan(\pi/4+x/2)|+c$$
  • $$x\sec x-\log |\tan(\pi/4+x)|+c$$
  • $$ x\sec x+\log|\tan x/2|+c$$
$$\displaystyle \int\{\frac{(\log x-1)}{1+(\log x)^{2}}\}^{2}dx$$ is equal to
  • $$\displaystyle \frac{x}{(\log x)^{2}+1}+c $$
  • $$\displaystyle \frac{xe^{x}}{1+x^{2}}+c$$
  • $$^{\dfrac{x}{x^{2}+1}+c} $$
  • $$\displaystyle \frac{\log x}{(\log x)^{2}+1}+c$$
$$\displaystyle \int \sec^{2}x.\text{cosec}^{2}xdx=$$
  • $$ \tan x-\cot x +c$$
  • $$\tan x +\cot x +c$$
  • $$-\tan x + \cot x +c$$
  • $$\sec x \tan x +c$$
$$\displaystyle \int\sqrt{\frac{x}{x-1}}dx, x\in(0,\pi/2)$$ equals
  • $$\sqrt{x(x-1)}+\log(\sqrt{x}+\sqrt{x-1})+c$$
  • $$\sqrt{x(x-1)}-\log(\sqrt{x}+\sqrt{x-1})+c$$
  • $$\sqrt{x(x-1)}+\log(\sqrt{x}-\sqrt{x-1})+c$$
  • $$\sqrt{x(x+1)}-\log(\sqrt{x}-\sqrt{x+1})+c$$
$$\displaystyle \int xe^{2x}(1+x)dx$$ equal to

  • $$\displaystyle \frac{xe^{x}}{2}+c$$
  • $$\displaystyle \frac{(e^{x})^{2}}{2}r$$
  • $$\displaystyle \frac{(1+x)^{2}}{2}+c$$
  • $$\displaystyle \frac{(xe^x)^2}{2}$$
The value of $$\displaystyle \int x^{2}2^{3x}dx=$$
  • $$\displaystyle \frac{x^{2}2^{3x}}{4\log 2}-\frac{x.2^{3x+1}}{9.(\log 2)^3}+\frac{2^{3x+1}}{27(\log 2)}+c$$
  • $$\displaystyle \frac{x^{2}2^{3x}}{3\log 2}-\frac{x.2^{3x+1}}{9.(\log 2)}+\frac{2^{3x+1}}{27(\log 2)}+c$$
  • $$\displaystyle \frac{x^{2}2^{3x}}{3\log 2}-\frac{2^{3x+1}.x}{3^{2}(\log 2)^{2}}+\frac{2^{3x+1}}{3^{3}(\log 2)^{3}}+c$$
  • $$\displaystyle \frac{x^{2}2^{3x}}{\log 2}-\frac{2^{3x+1}.x}{3^{2}(\log 2)^{2}}+\frac{2^{3x+1}}{(\log 2)^{3}}+c$$
$$\displaystyle \int\log{x}d{x}=$$
  • $$x(\log x-x)+c$$
  • $$ x(\log x)+c$$
  • $$ x(\log x-1)+c$$
  • $$ x(\log x+x)+c$$
Evaluate : $$\displaystyle \int\frac{\cot^{2}x}{(co\sec^{2}x+co\sec x)}d{x}$$
  • $$ x-\sin x +C$$
  • $$x + \cos x +C$$
  • $$ \sin x - x+C  $$
  • $$2\tan(\displaystyle \frac{ax}{2})+c$$
$$\displaystyle \int[\frac{cosx}{x}-\sin{x}\log x]dx=$$
  • $$ (\log x) sinx+c$$
  • $$(\log x) (cos x) +c$$
  • $$2(\log x) (cos x) +c$$
  • $$-(\log x) sin x+c$$
Evaluate $$\displaystyle \int x^{2}e^{x}dx=$$
  • $$ e^{x}(x^{2}-2x+2)+c$$
  • $$ e^{x}(x^{2}+2x+2)+c$$
  • $$ x^{2}+ex+c$$
  • $$ e^{x}(x^{2}+x+2)+c$$
$$\displaystyle \int e^{\log x}\cos xdx=$$
  • $$ xsinx-cosx+c$$
  • $$\dfrac{x}{2}sinx+cos^{2}x+c$$
  • $$ xsinx+cosx+c$$
  • $$ xsinx+sinx+c$$
$$\displaystyle \int\frac{x\cos x\log x-\sin x}{x(\log x)^{2}}dx$$ is equal to
  • $$\displaystyle \frac{\sin x}{\log x}+c$$
  • $$ \log{x}+\sin{x+c}$$
  • $$ \log{x}\sin{x+c}$$
  • $$\displaystyle \frac{\cos x}{\log x}+c$$
The value of $$\int x(cosecx\ cotx)dx=$$ is
  • $$ xcosec x-log |tanx/2|+c$$
  • $$2-xcosec x+\log| tan \displaystyle \frac{x}{2}|+c$$
  • $$ xcosec x-2 \log|tanx/2|+c$$
  • $$ xcot x-log |tan{\displaystyle \frac{x}{2}}|+c$$
Solve  $$\displaystyle \int \tan^{-1}xdx$$
  • $$ x\tan x -\log|1+x^{2}|+c$$
  • $$x$$ $$\tan^{-1} x -\displaystyle \frac{1}{2}\log|1+x^{2}|+c$$
  • $$ x$$  $$\tan^{-1} x+\log\sqrt{1+x^{2}}+c$$
  • $$ x\tan x+\log|1+x^{2}|+c$$
Evaluate $$\displaystyle \int\frac{\log(x/e)}{(\log x)^{2}}dx$$
  • $$\displaystyle \frac{\log x}{x}+c$$
  • $$ \displaystyle \frac{x}{\log x}+c$$
  • $$^{\dfrac{x}{log(x)^{2}}+c} $$
  • $$\displaystyle \frac{(\log x)^{2}}{x}+c$$
$$\displaystyle \int x^{2} {\it cosh} 4xdx=$$
  • $$\displaystyle \frac{x^{2}}{4} {\it sinh}4x-\displaystyle \frac{x}{8}{\it cosh} 4x+\displaystyle \frac{\sinh 4x}{32}+c$$
  • $$\displaystyle \frac{x^{2}}{4}sinh4x+\frac{x}{8}$$ cosh $$4{x-}\displaystyle \frac{\sinh 4x}{32}+c$$
  • $$\displaystyle \frac{x^{2}}{4}sinh4x-\frac{x}{8}$$ cosh $$4x-\displaystyle \frac{\sinh 4x}{32}+c $$
  • $$\displaystyle \frac{x^{2}}{4}sinh4x+\frac{x}{8}$$ cosh $$4x-\displaystyle \frac{\sin h4x}{64}+$$
$$\displaystyle \int cot^{-1}(\frac{x-1}{x+1})dx=$$
  • $$ \displaystyle \frac{\pi}{4}x+x\cot^{-1}x-\frac{1}{2}\log(1+x^{2})+c$$
  • $$ x\displaystyle \cot^{-1}x+\frac{1}{2}\log(1+x^{2})+c$$
  • $$\displaystyle \frac{\pi}{4}x-\frac{1}{2}\log(1+x^{2})+c$$
  • $$^{\displaystyle \frac{\pi}{4}x+x\cot^{-1}x}+\displaystyle \frac{1}{2}\log(1+x^{2})+c$$
$$\displaystyle \int e^{x}cosec x(1-\cot x)dx=$$
  • $$ e^{x}\cot x+c$$
  • $$ e^{x}cosec x\cot x+c$$
  • $$ e^{x}cosec x+c$$
  • $$-e^{x}\cot x+c$$
$$\displaystyle \int x^{2}a^{x}dx=$$
  • $$ a^{x}[\displaystyle \frac{x^{2}}{\log a}-\frac{2x}{(\log a)^{2}}+\frac{2}{(\log a)^{3}}]+c$$
  • $$ a^{x}[\displaystyle \frac{x^{2}}{\log a}+\frac{2x}{(\log a)^{2}}-\frac{2}{(\log a)^{3}}]+c$$
  • $$ a^{x}[\displaystyle \frac{x^{2}}{\log a}+\frac{2x}{(\log a)^{2}}+\frac{2}{(\log a)^{3}}]+c$$
  • $$ a^{x}(\displaystyle \frac{x}{\log a}-\frac{1}{(\log a)^{2}})+c$$
lf polynomials $$P$$ and $$Q$$ satisfy $$\displaystyle{\int((3x-1)\cos x+(1-2x)\sin x)dv}=P\cos x+Q\sin x+R$$ (ignoring the constant of integration) then
  • $$ P=3x-2$$
  • $$ Q=2+x$$
  • $$ P=3(x-1)$$
  • $$Q=3(x-1)$$
Evaluate $$\displaystyle \int\frac{\arcsin\sqrt{x}}{\sqrt{1-x}}dx$$ $$=$$
  • $$2[\sqrt{x}-\sqrt{1-x}$$ arc $$\sin\sqrt{x}]+c$$
  • $$2[\sqrt{x}+\sqrt{1-x}$$ arc $$\sin\sqrt{x}]+c$$
  • $$2[\sqrt{x}+\sqrt{1-x}$$ arc $$\cos\sqrt{x}]+c$$
  • $$2[\sqrt{x}-\sqrt{1-x}$$ arc $$\cos\sqrt{x}]+c$$
$$\displaystyle \int e^{x}(x^{2}-5x+8)$$ dx $$=e^{x}f(x)+c$$ then $$f(x)$$
  • $$x^{2}-5x+12$$
  • $$ x^{2}+7x+15$$
  • $$ x^{2}-7x-15$$
  • $$ x^{2}-7x+15$$

Evaluate $$\displaystyle \int x\frac{(\sec 2x-1)}{(\sec 2x+1)}dx$$
  • $$x \tan x-\log |\displaystyle \sec x|+\frac{x^{2}}{2}+c$$
  • $$x \tan x-\log |\displaystyle \sec x|-\frac{x^{2}}{2}+c$$
  • $$x \tan x-\log |\sec\frac{x}{2}|+c$$
  • $$x \tan x-\log |\displaystyle \sec\frac{x}{2}|+\frac{x^{2}}{2}+c$$
$$\displaystyle \int e^{x}2^{3\log_{2}x}dx=e^{x}f(x)+c$$, then $$f(x)=$$
  • $${ x }^{ 3 }-3{ x }^{ 2 }+6x-6$$
  • $$x^{3}-3x^{2}-6x-3$$
  • $$ x^{3}-3x^{2}+6x+6$$
  • $$x^{3}+3x^{2}+6x+6$$
$$\displaystyle \int\frac{x+\sin x}{1+\cos x}dx=$$
  • $$xtan \displaystyle \frac{x}{2}+c$$
  • $$xcot \displaystyle \frac{x}{2}+c$$
  • $$xsin \displaystyle \frac{x}{2}+c$$
  • $$xcos \displaystyle \frac{x}{2}+c$$
$$\displaystyle \int\cos^{-1}(\frac{1}{x})dx$$ equal to
  • $$ x\sec^{-1}x+\cosh^{-1}x+c$$
  • $$ x\sec^{-1}x-\cosh^{-1}x+c$$
  • $$ x\sec^{-1}x-\sin^{-1}x+c$$
  • $$ x\sec^{-1}x+\sin^{-1}x+c$$
Evaluate: $$\displaystyle \int\log_{10}xdx$$
  • $$ (x-1)\log_{e}x+c$$
  • $$ \displaystyle \log_{e}10.x\log_{e}(\frac{x}{e})+c$$
  • $$ \displaystyle \log_{10}e.x\log_{e}(\frac{x}{e})+c$$
  • $$\cfrac{1}{x}+c$$
Evaluate $$\displaystyle \int\frac{xe^{x}}{(x+1)^{2}}dx$$
  • $$ \displaystyle e^{x}[\dfrac{1}{x+1}]+c$$
  • $$ \displaystyle e^{x}[\dfrac{-1}{x+1}]+c$$
  • $$ e^{x}x+c$$
  • $$\displaystyle e^{x}[\dfrac{-1}{(x+1)^{2}}]+c$$
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