INDEFINITE INTEGRAL

 $\int_{ }^{ }\frac{x^2-1}{\left(x^4+3x^2+1\right)Tan^{-1}\left(\frac{x^2+1}{x}\right)}\$  dx is equal to

•  $Tan^{-1}\left(x+\frac{1}{x}\right)+c$  

•  $\log_e\left(Tan^{-1}\left(x+\frac{1}{x}\right)\right)+c$  

•  $\log_e\left(\tan\left(\frac{x^2+1}{x}\right)\right)+c$  

•  $\left(x+\frac{1}{x}\right)Tan^{-1}\left(x+\frac{1}{x}\right)+c$  

The value of the integral $\int_{ }^{ }\frac{\cos^3x+\cos^5x}{\sin^2x+\sin^4x}dx\ is$  

•  $\sin x-6Tab^{-1}\left(\sin\ x\right)+c$  

•  $\sin x-2\left(\sin x\right)^{-1}+c$  

•  $\sin x-2\left(\sin x\right)^{-1}-6Tan^{-1}\left(\sin x\right)+c$  

•  $\sin x-2\left(\sin x\right)^{-1}+5Tan^{-1}\left(\sin x\right)+c$  

 $\int_{ }^{ }\frac{x+1}{x\left(1+xe^x\right)^2}dx=\log_e\left|\frac{xe^x}{1+xe^x}\right|+f\left(x\right)+c,$  then f(x) is.

•  $\frac{1}{1+xe^x}$  

•  $\frac{x}{1+xe^x}$  

•  $\frac{xe^x}{1+x}$  

•  $\frac{xe^x}{1+e^x}$  

 $\int_{ }^{ }\frac{x-1}{\left(x+1\right)\sqrt{x\left(x^2+x+1\right)}}dx\ is$  

•  $Tan^{-1}\left(\frac{x^2+x+1}{x}\right)+c$  

•  $2Tan^{-1}\left(\frac{x^2+x+1}{x}\right)+c$  

•  $Tan^{-1}\left(\frac{\sqrt{x^2+x+1}}{x}\right)+c$  

•  $2Tan^{-1}\sqrt{x+\frac{1}{x}+1}+c$  

 $\int_{ }^{ }\frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4}dx=$  

•  $\frac{3}{8}\left(\frac{1}{x^2}-1\right)^{\frac{4}{3}}+c$  

•  $-\frac{3}{8}\left(\frac{1}{x^2}+1\right)^{\frac{4}{3}}+c$  

•  $-\frac{3}{8}\left(\frac{1}{x^2}-1\right)^{\frac{4}{3}}+c$  

•  $-\frac{3}{4}\left(1-\frac{1}{x^2}\right)^{\frac{4}{3}}+c$  

 $If\ y\left(x-y\right)^2=x,\ then\ \int_{ }^{ }\frac{dx}{x-3y}equals$  

•  $\frac{x}{2}\log_e\left\{\left(x-y\right)^2+-1\right\}+c$  

•  $\frac{1}{2}\log_e\left\{\left(x-y\right)^2-1\right\}+c$  

•  $x+\frac{1}{2}\log_e\left\{\left(x-y\right)^2+1\right\}+c$  

•  $\log_e\left\{\left(x-y\right)^2-1\right\}+c$  

 $If\ \int_{ }^{ }\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)^{\frac{1}{2}}\ \frac{dx}{x}=2\ \cos^{-1}\sqrt{x}-\phi\left(x\right)+c$  , then  $\phi\left(x\right)\ equals$  

•  $\log_e\left(\frac{1-\sqrt{1-x}}{\sqrt{x}}\right)$  

•  $\frac{1}{2}\log_e\left(\frac{1+\sqrt{1-x}}{\sqrt{x}}\right)$  

•  $2\log_e\left(\frac{1-\sqrt{1-x}}{\sqrt{x}}\right)$  

•  $2\log_e\left(\frac{1+\sqrt{1-x}}{\sqrt{x}}\right)$  

 $Let\ I_1=\int_{ }^{ }\frac{x^1+1}{\left(x^3+3x+1\right)^{\frac{1}{3}}}dx=\frac{1}{2}\left(x^3+3x+1\right)^{\frac{2}{3}}+c$  and  $I_2=\int_{ }^{ }\frac{\sin2x}{1+\sin^2x}dx=\log_e\left(1+\sin^2x\right)+c$  Then

•  $both\ I_1and\ I_2\ are\ correct$  

•  $both\ I_1\ and\ I_2\ are\ not\ correct$  

•  $I_1is\ correct\ and\ I_2\ is\ not$  

•  $I_2\ is\ correct\ and\ I_1\ is\ not$  

 $\int_{ }^{ }\frac{dx}{\sqrt{1+x^2}\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)}}is$  

•  $2\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}$  

•  $\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}$  

•  $\frac{1}{2}\log_e\left(x+\sqrt{1+x^2}\right)+c$  

•  $\frac{1}{2}\sqrt{\log_e\left(x+\sqrt{1+x^2}\right)+c}$  

 $\int_{ }^{ }\frac{\tan x}{1+\tan x+\tan^2x}dx\ is\ equal\ to$  

•  $x+\frac{2}{\sqrt{3}}Tan^{-1}\ \frac{\left(2\tan x+1\right)}{\sqrt{3}}+c$  

•  $x-\frac{2}{\sqrt{3}}Tan^{-1}\left(\frac{2\tan x+1}{\sqrt{3}}\right)+c$  

•  $x+\frac{1}{\sqrt{3}}Tan^{-1}\left(\frac{\tan x+1}{\sqrt{3}}\right)+c$  

•  $x-\frac{1}{\sqrt{3}}Tan^{-1}\left(\frac{\tan x+1}{\sqrt{3}}\right)+c$