Quiz 1 Calculus 2

Evaluate the  $\int\ \frac{dx}{1+\sqrt{x}}$  .

•  $\arctan\sqrt{x}\ +C$  

•  $2\left(1+\sqrt{x}\right)-2\ln\left|1+\sqrt{x}\right|\ +C$  

•  $\ln\left|1+\sqrt{x}\right|+C$  

•  $2\sqrt{x}-2\ln\left|1+\sqrt{x}\right|\ +C$  

•  $\ln\left|1+\sqrt{x}\right|-\arctan\sqrt{x}\ +C$  

Find the  $\int\ \frac{2x+3}{\sqrt{x+1}}dx$  .

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

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

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

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

•  $2\sqrt{x+1}+\frac{4}{3}\left(x+1\right)\sqrt{x+1}+C$  

Evaluate the  $\int\ \frac{\sqrt{x^2-1}}{x}dx$  .

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

•  $\sqrt{x^2-1}\ -\arctan x\ +C$  

•  $\operatorname{arcsec}\ x-\sqrt{x^2-1}\ +C$  

•  $\arcsin\left(x^2-1\right)\ +\sqrt{x^2-1}\ +C$  

•  $\sqrt{x^2-1}-\operatorname{arcsec}x\ +C$  

Find the  $\int\ \frac{e^{\frac{1}{x^2}}}{x^3}dx$  .

•  $\frac{e^{\frac{1}{x}}}{x}\ +e^x+C$  

•  $-e^{x^{-2}}+C$  

•  $e^{\frac{1}{x^2}}-\frac{e^{\frac{1}{x^2}}}{x}+C$  

•  $e^{-x^2}+x^2+C$  

•  $e^{x^{-2}}+C$  

Evaluate the  $\int\ \frac{\ln x}{\left(x+1\right)^3}dx$  .

•  $\frac{1}{3}\ln\left|x+1\right|-\frac{\ln x}{x+1}-\frac{1}{3}\ln\ x\ +C$  

•  $\frac{\ln\ x}{x+1}-\frac{1}{3}\ln x-\frac{1}{3}\ln\left|x+1\right|+C$  

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

•  $\frac{1}{3}\ln\left|x+1\right|-3\ln\ X-\frac{\ln\ x}{\left(x+1\right)^2}+C$  

•  $\frac{\ln x}{\left(x+1\right)^3}-\frac{1}{3}\ln\left|x+1\right|+\ln\ x\ +C$  

Find the  $\int\ \frac{x^3}{x^2+2x+5}dx$  .

•  $\ln\sqrt{x^2+2x+5}\ +x^2-2x\ -\ \frac{1}{2}\arctan\left(\frac{x+1}{2}\right)+C$  

•  $5\arctan\left(\frac{x}{5}\right)+\ln\sqrt{x^2+2x+5}\ +\frac{x^2}{2}-2x+C$  

•  $5\arctan\left(x+2\right)-\ln\sqrt{x^2+2x+5}+x-2+C$  

•  $\frac{11}{2}\arctan\left(\frac{x+1}{2}\right)+\frac{\left(x-2\right)^2}{2}-\ln\sqrt{x^2+2x+5}\ +C$  

•  $x-2+\frac{11}{2}\arctan\left(\frac{x+1}{2}\right)-\ln\sqrt{x^2+2x+5}\ +C$  

Evaluate the  $\int\ \frac{2x+3}{x^2+2x+2}dx$  .

•  $\ln\left|x^2+2x+2\right|\ +C$  

•  $\arctan\left(x+1\right)+\ln\left|x^2+2x+2\right|+C$  

•  $2\ln\left|x^2+2x+2\right|+3\arctan\left(x+1\right)+C$  

•  $\ln\left|x+1\right|+\ln\left|x+2\right|+C$  

•  $\ln\left|x^2+2x+2\right|+\frac{3}{2}\arctan\left(\frac{x+1}{2}\right)+C$  

Evaluate the  $\int\ \frac{\sin^5\theta}{\cos\theta}d\theta$  .

•  $\ln\left|\cos\theta\right|+\frac{1}{2}\sin^2\theta+\frac{\sin^5\theta}{5}+C$  

•  $\frac{1}{2}\cos^2\theta-\frac{1}{4}\cos^4\theta+\ln\left|\sec\theta\right|+C$  

•  $\ln\left|\sec\theta\right|\ \ \ +\ \cos^2\theta-\frac{1}{4}\cos^4\theta\ +C$  

•  $\tan\theta-\cos^2\theta+\frac{\cos^4\theta}{4}+C$  

•  $\ln\left|\sec\theta\right|-\frac{1}{2}\sin^2\theta+\frac{\sin^4\theta}{4}+C$  

Evaluate the  $\int\ \sin^3\theta\sec^4\theta\ d\theta$  .

•  $\frac{1}{3}\sin^3\theta-\ln\left|\sec\theta\right|+C$  

•  $\frac{1}{3}\sec^3\theta-\sec\theta+C$  
  

•  $\frac{1}{4}\sin^4\theta+\frac{1}{3}\sec^3\theta+C$  
  

•  $\frac{1}{5}\sec^5\theta+\frac{1}{4}\sin^4\theta+C$  
  

•  $\frac{1}{7}\cos^7\theta-\frac{3}{5}\cos^5\theta+\frac{1}{3}\cos^3\theta+C$