$\int_{ }^{ }\frac{x^3}{x+1}dx=$  

•  $\frac{x^3}{3}-\frac{x^2}{2}+x-\ln\left|x+1\right|+c$  

•  $\frac{x^3}{3}-2x+1-\ln\left|x\right|+c$  

•  $3x^2-\frac{x^2}{2}+x-\ln\left|x+1\right|+c$  

• none of these

 $\int_{ }^{ }\frac{x^3}{2x+1}dx=$  

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

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

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

• None

 $\int_{ }^{ }\left(e^{x\ln a}-e^{a\ln x}+3e^{a\ln a}\right)dx=$  

•  $\frac{a^x}{\ln a}-\frac{x^{a+1}}{a+1}+3x+c$  

•  $\frac{a^x}{\ln a}+\frac{x^a}{a+1}+3+c$  

•  $\frac{a^x}{\ln a}+\frac{x^{a+1}}{a}+c$  

• None

 $\int_{ }^{ }\frac{1}{\sqrt{1-4x^2}}dx=$  

•  $\frac{\sin^{-1}x}{2}+c$  

•  $\frac{\sin^{-1}2x}{2}+c$  

•  $\sin^{-1}2x+c$  

•  $\frac{\sin^{-1}2x}{3}+c$  

 $\int_{ }^{ }\sin^2x.\cos^2xdx=$  

•  $\frac{1}{8}\left[x+\frac{\sin4x}{4}\right]+c$  

•  $\frac{1}{8}\left[x-\frac{\sin4x}{4}\right]+c$  

•  $\frac{1}{32}\left[x-\frac{\sin4x}{4}\right]+c$  

• None of these

 $\int_{ }^{ }\left(\frac{\sin^6x+\cos^6x}{\tan^2x}\right)\sec^4xdx=$  

• tanx+cotx+c

• cotx-tanx+c

• tanx-cotx+c

• None of these

If $\int_{ }^{ }\frac{\sin x}{\sin\left(x-\alpha\right)}dx\ =$   $Ax+B\log\sin\left(x-\alpha\right)+C\$   $then\ \left(A,B\right)=$  

•  $\left(\cos\alpha,\sin\alpha\right)$  

•  $\left(\sin\alpha,\cos\alpha\right)$  

•  $\left(\sin\alpha,-\cos\alpha\right)$  

•  $\left(-\cos\alpha,-\sin\alpha\right)$  

The value of $\sqrt{2}\int_{ }^{ }\frac{\sin xdx}{\sin\left(x-\frac{\pi}{4}\right)}=$  

•  $x+\ln\left|\sin\left(x-\frac{\pi}{4}\right)\right|+c$  

•  $2\left[x+\ln\left|\sin\left(x-\frac{\pi}{4}\right)\right|\right]+c$  

•  $2\left[x-\ln\left|\sin\left(x-\frac{\pi}{4}\right)\right|\right]+c$  

• None of these

 $\int_{ }^{ }\frac{dx}{\cos x-\sin x}=$  

•  $\frac{1}{\sqrt{2}}\ln\left|\tan\left(\frac{x}{2}+\frac{3\pi}{8}\right)\right|+c$  

•  $\sqrt{2}\ln\left|\tan\left(\frac{x}{2}+\frac{3\pi}{8}\right)\right|+c$  

•  $\sqrt{2}\ln\left|\tan\left(\frac{x}{2}-\frac{3\pi}{8}\right)\right|+c$  

• None of these

 $\int_{ }^{ }\frac{dx}{\cos x+\sqrt{3}\sin x}=$  

•  $\frac{1}{2}\ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)\right|+c$  

•  $\frac{1}{2}\ln\left|\tan\left(\frac{x}{2}-\frac{\pi}{12}\right)\right|+c$  

•  $\frac{1}{2}\left[\ln\left|\tan\left(x-\frac{\pi}{12}\right)\right|\right]+c$  

• None