Calculus 2 Review of Methods of Integration

 $\int_{ }^{ }\left(2-e^{3x}\right)^2\ dx$  What method below would you use to rewrite the given integral?  

• Integration By Parts
  

• Add and subtract terms

• Expand the expression

• Multiply by complex conjugate

Check all the methods or integration rules you would use to integrate the following integral:  $\int_{ }\ \frac{1-\cos\left(2x\right)}{\sin^2\left(2x\right)}dx$  

• Using trig identities

• multiply by complex conjugate

• Separate the numerator

• use basic trig integrals

• use the power rule when n is not equal to -1

Would you use the Tabular Method to integrate  $\int\ \frac{1}{3}x^2\cdot\ln\left(4x\right)dx$  ?

• YES

• NO

When integrating the integral  $\int\ \frac{1}{3}x^2\cdot\ln\left(4x\right)dx$ , the solution will be.....

•  $F\left(x\right)=x^3\ln4x-\frac{1}{3}x^3+c$  

•  $F\left(x\right)=\frac{1}{3}x^3\cdot\ln4x-\frac{1}{9}x^3+c$  

•  $F\left(x\right)=\frac{1}{9}x^3\cdot\ln\left(4x\right)-\frac{1}{27}x^3+c$  

•  $F\left(x\right)=\frac{1}{9}x^3\cdot\ln x-x^2+c$  

Which integral is an "engineering problem" when using integration by parts?

• $\int\ 2x\cdot\cos\left(x^2\right)dx$

• $\int_{ }2x^3\cdot\cos4xdx$

• $\int\ 2x^3\cdot e^{-x}dx$

• $\int\ e^{-x}\cdot\cos4xdx$

What integral rule on your formula sheet would you use for  $\int-3\csc^2\left(5x\right)\cdot e^{\cot\left(5x\right)}dx$  ?

• #14  $\int\csc^2udu=-\cot u+c$  

• #20  $\int a^udu=\frac{1}{\ln a}a^u+c$  

• #16  $\int\csc u\cdot\cot udu=-\csc u+c$  

• #6  $\int e^udu=e^u+c$  

What basic integral rule would you use to integrate the following:  $\int\ 5^{\sin3x}\cdot\cos3x\ dx$  

•  $\int U^ndu=\frac{U^{\left(n+1\right)}}{n+1}+c$  

•  $\int\cos u\ du=\sin u\ +c$  

•  $\int a^udu=\frac{1}{\ln a}a^u+c$  

The derivative of  $y=\frac{1}{2}e^{\tan3x}$  is which of the following?

•  $y'=\frac{1}{2}e^{\tan3x}$  

•  $y'=\frac{3}{2}e^{\tan3x}$  

•  $y'=\frac{1}{2}e^{\tan3x}\sec^23x$  

•  $y'=\frac{3}{2}e^{\tan3x}\sec^23x$  

How would you rewrite the function  $y=\ln\left(\frac{\sqrt{e^x}}{\cos4x}\right)$  using the laws of logs ?

•  $y=\frac{1}{2}\ln e^x-\ln\left(\cos4x\right)$  

•  $y=\frac{1}{2}\ln e^x+\ln\left(\cos4x\right)$  

•  $y=\ln\sqrt{e^x}\cdot\ln\left(\cos4x\right)$  

If you integrate the following integral $\int\left(4x^2-5x\right)e^{-3x}dx$  using the Tabular method, then the first term in your answer will be ?

•  $-\left(4x^2-5x\right)e^{-3x}$  

•  $-\frac{1}{3}\left(4x^2-5x\right)e^{-3x}$  

•  $-3\left(4x^2-5x\right)e^{-3x}$  

•  $\left(4x^2-5x\right)e^{-3x}$