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HONORS CALC II F 08, MIDTERM I
(1) Find the derivative of the function Zx
√x
et
tdt.
(Suggestion: Write the integral as a difference between two integrals, one from 0 to xand
another from 0 to √x.
Write f(t) = et
t. Set
I(x) = Zx
√x
f(t)dt =I1(x)−I2(x),where I1(x) = Zx
0
f(t)dt, I2(t) = Z√x
0
f(t)dt.
Hence by FTC and the chain rule,
I0(x) = f(x)−
f(√x)
2√x=ex
x−
e√x
2x=2ex−e√x
2x.
(2) Evaluate the integral
Zdx
x2√x2
−16.
Change x= 4 sec t. Then dx = 4 tan tsec tdt and x2
−16 = 16 tan2t. So
I=Ztan tsec t
16 sec2ttan tdt =1
16 Zcos tdt =sin t
16 +C,
Now t= arccos(4/x), so sin t=√1−cos2t, and we have
I=p1−16/x2
16 +C=√x2
−16
16x+C.
(3) Evaluate the integral
Zdx
√1 + x+√x.
(You don’t need to change variables, you can do algebraic manipulation using the identity
(a+b)(a−b) = a2
−b2).
Following the suggestion, we note that
1
√1 + x+√x=1
√1 + x+√x
√1 + x−√x
√1 + x−√x=√1 + x−√x.
Therefore
I=2
3(1 + x)3/2
−x3/2+C.
(4) Evaluate the improper integral
Z100
0
√tln tdt.
Integrate by parts
Z100
0
√tln tdt =2
3t3/2ln t
100
0−
2
3Z100
0
t1/2dt =2000
3ln 100 −
4000
9=4000
3ln 10 −
1
3.
1
