Applied Mathematics 30063
Exercise 1 Integral calculus
Multiple choices questions (5 points each)
1. An antiderivative of f(x) = 3
px1on the interval I= (0;10) is:
A3
4(x1) 3
px1B4
3(x1) 3
px1C4
3
4
q(x1)3
Ddoes not exist E1
33
q(x1)2
Fnone of the preceding
2. If Rb
0x2dx =8
3then b=
A2B2C2
3
D2
3E2Fnone of the preceding
3. The antiderivative of f(x) = 2xpassing through the point (x0; y0) = (0;3) is:
A2x
ln 2 + 3 B2x
ln 2 + 3 1
ln 2 C2x
ln 2 + 3 1
ln 2
D2x
ln 2 +1
ln 2 E2x
ln 2 +3+ 1
ln 2 Fnone of the preceding
4. G(x)is the integral function of f(x) = 4x3x2with initial point a= 3. Then G(x) =
A8x9x2B8x9x2+ 67 Cx3+ 2x2+ 9
D2x2x3E2x2x3+ 3 Fnone of the preceding
5. If f(x) = ex
(ex+ 1)2then R+1
1 f(x)dx:
Adiverges to +1Bdiverges to 1 Cconverges to 0
Dconverges to < 0Econverges to > 0Fnone of the preceding
Short answer questions (0 to 5 points each)
1. Let fbe an integrable function on Rand let Gbe an antiderivative of fon R. Then it is:
(a) Rf(x)dx =_______________ (b) Rb
af(x)dx =______________ (c) Rb
1 f(x)dx =_____________
2. Let f:R!Rbe everywhere di¤erentiable. Using the substitution y=f(x), it is
Zf0(x)[f(x)]2dx=________________________________________
3
