PROF. A. SUCIU

MTH 1108 -- Foundations of Integral Calculus -- Spring 1999

Answers to Quiz 3

Problem 1

Part a.

Integrate[Sqrt[x]+1/(2*x),x]
        3/2
     2 x
     ------ + Log[Sqrt[x]]
       3

Integrate[Sqrt[x]+1/(2*x),{x,4,9}]
     38
     -- - Log[2] + Log[3]
     3

N[%]
     13.0721

Part b.

Integrate[x*Exp[x^2],x]
       2
      x
     E
     ---
      2

Integrate[x*Exp[x^2],{x,0,Sqrt[2]}]
             2
       1    E
     -(-) + --
       2    2

N[%]
     3.19453

Part c.

Factor[Integrate[(2*x+1)/(x^2+x)^3,x]]
          -1
     -------------
        2        2
     2 x  (1 + x)

Integrate[(2*x+1)/(x^2+x)^3,{x,1,2}]
     1
     -
     9

N[%]
     0.111111

Problem 2

FilledPlot[(2*x+3)^2,{x,-2,3},AspectRatio->1]
[Graphics:1108.ansq3gr2.gif][Graphics:1108.ansq3gr1.gif] 

Integrate[(2*x+3)^2,x]
                     3
              2   4 x
     9 x + 6 x  + ----
                   3

Integrate[(2*x+3)^2,{x,-2,3}]
     365
     ---
      3

N[%]
     121.667

Problem 3

FilledPlot[{4*x-3,x^2},{x,-0.5,3.5},AspectRatio->0.8]
[Graphics:1108.ansq3gr2.gif][Graphics:1108.ansq3gr3.gif] 

Solve[4*x-3==x^2,x]
     {{x -> 1}, {x -> 3}}

Integrate[4*x-3-x^2,x]
                    3
               2   x
     -3 x + 2 x  - --
                   3

Integrate[4*x-3-x^2,{x,1,3}]
     4
     -
     3

N[%]
     1.33333

Problem 4

f[x_]:=D[F[x],x]; F[1]:=2; F[3]:=4;

Integrate[f[x],{x,1,3}]
     2

Integrate[f[x],{x,0,6}]+Integrate[f[x],{x,6,0}]
     0

Integrate[2*f[x],{x,1,3}]-Integrate[5*f[x],{x,1,3}]
     -6

Integrate[f[x],{x,1,2}]+Integrate[f[x],{x,2,3}]
     2

Integrate[Sqrt[f[x]],{x,3,3}]
     0

Problem 5

g[x_]:=1/(1+Sqrt[x])

G[x_]:=Integral[g[t],{t,2,x^2}]

dG[x_]=g[x^2]*D[x^2,x]
         2 x
     ------------
               2
     1 + Sqrt[x ]

dG[9]
     9
     -
     5

N[%]
     1.8