Math 205 Midterm Test
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1. (8 marks)
a) Evaluate the intgral R 1
−3
|x| dx by interpreting it
in terms
of area.
b) Find the derivative dF /dx
of the function
0
F (x) =
Z
x2 −1
sin(t + 1)
dt t
+ 1
2. (5 marks) Find the antiderivative F
(x) of the
function
f (x)
= x e−x2
such that F (0) = 3.
3.
(12 marks) Calculate the following indefinite integrals
(a)
Z (√2x − 1)2
x
dx (b)
Z
4t2 ln(t) dt (c)
Z x − 1 dx x2 − 7x + 12
4. (8 marks) Evaluate the following definite integrals (do not approxi-
mate ):
(a)
3
Z 1 + arctan(x/3)
9 + x2
0
dx (b)
1
Z
x e−xdx
0
5. (7 marks) Find the average value of f (x) = 1
+
sin2(x)
on the interval [0, π].
Bonus Question
(2 marks) Calculate
the definite
integral
2
Z
|
0
in terms of area (HINT:
sketch the function)
1
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