Final Exam

1. (35%) Evaluate the following integrals. You must show all your work.

   1. $$\int x^{2} \sin(x^{3}+1) \, dx$$

   2. $$\int e^{\sqrt{2} x + 3} \, dx$$

   3. $$\int x^{5} \ln x \, dx$$

   4. $$\int x e^{3x} \, d x$$

   5. $$\int \frac{1}{x^{2} \sqrt{x^{2} - 9}} \, d x$$
      Hint: let $x=3 \sec \theta$ .

   6. $$\int_{-\infty}^{\infty} \frac{1}{x^{2}+25} \, dx$$

   7. $$\int_{0}^{\infty} x e^{-x^{2}} \, dx$$

2. (10 %) Find the area bounded by the curves $y=6-3x$ , $y=x^2-4$ , and the $y$ axis.
   

3. (10 %) Find the volume of revolution bounded by $y=\sqrt{x+3}$ , $y=0$ , $x=-3$ , and $x=3$ .
   

4. (10%) What is the arc length of the curve $y = 20 \cosh (x/20) -15$ for $-7 \le x \le 7$ ?
   Hint: $(\cosh u)^{\prime}=\sinh u$ , $(\sinh u)^{\prime}=\cosh u$ , $\cosh^{2} u - \sinh^{2} u = 1$ .
5. (10%) Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$ . Show all your work.
   1.      $e^{x} \sin x$ ,     $a=0$ ,     $n=3$
   2.      $(1-3x)^{-1/2}$ ,     $a=0$ ,     $n=3$

6. (10%) Estimate the value of $${\int_{0}^{1/2} \frac{1}{1+x^{6}}\, dx}$$ by approximating the integrand with a Taylor polynomial of degree $6$ .

7. (20%) Find the solutions to the following differential equations.

   1. $$\frac{d y}{d x} = 0.5 (y - 200), \quad y(0) = 50 .$$

   2. $$y^{\prime\prime} - 3 y^{\prime} - 4 y = 0, \quad y(0)=1, \ y^{\prime}(0) = 0 .$$