Math 2 - Review Problems

1. Find $${\int (2 \pi x^{\pi -1} - 10 x^{3/2}) dx}$$. Answer: $${2 x^\pi - 4 x ^{5/2} + C}$$
   
   

2. Find $${\int( - \frac{4}{x} + 2 e^{4x}) dx}$$. Answer: $${-4 \ln \vert x\vert + \frac{1}{2}e^{4x}+C}$$
   
   

3. Find $${\int (e^{2x} + 4)^3 e^{2x} dx}$$. Answer: $${\frac{1}{8}(e^{2x}+4)^4+C}$$
   
   

4. Find $${\int \frac{1}{x(\ln x)^2} dx}$$. Answer: $${-(\ln x)^{-1}+ C}$$
   
   

5. Find $${\int \frac{x^3+2x}{x^2-1} dx}$$. Answer: $${\frac{1}{2}x^2+ \frac{3}{2} \ln \vert x^2-1\vert + C}$$
   
   

6. Find $${\int \frac{\ln x}{x^{3/2}} dx}$$. Answer: $${-2 x^{-1/2} \ln x - 4 x^{-1/2} + C}$$
   
   

7. Find $${\int x e^{-3x} dx}$$. Answer: $${ -\frac{1}{3} x e^{-3x} - \frac{1}{9} e^{-3x} + C}$$
   
   

8. Find $${ \int \frac{3x+2}{x^2-x-12} dx}$$. Answer: $${2 \ln \vert x-4\vert + \ln \vert x+3\vert + C}$$
   
   

9. Find $${\int \frac{2 x^2 - 9 x - 12}{x^3+x^2-12x} dx}$$. Answer: $${ \ln \vert x\vert + 2 \ln \vert x+4\vert - \ln \vert x-3\vert + C}$$
   
   

10. Find the general solution to the differential equation $${\frac{dy}{dx}= 6 x^2 \sqrt y}$$.

    Answer: $${y= (x^3+C)^2}$$
    
    

11. Sketch the graph of $f$ on the interval $[-2,4]$ and then evaluate $${\int_{-2}^4 f(x) dx}$$, where
    $$f(x)=\left\{\begin{array}{cc} -x & -2\le x\le 2 -2 & 2<x \le 4 \end{array} \right.$$
    Answer: $-4$
    
    

12. Evaluate $${\int_0^6 \left( \frac{1}{2}x^2 -4 \right) dx}$$. Answer: $12$.