Quiz No.8 - Boston University

Summer Term I
Kostadinov
MA124 Calculus II
Boston University
Quiz No.8
student:
Problem 1: Evaluate the definite integral
1
Z
e3x dx
0
Problem 2: Evaluate the indefinite integral
Z
xex dx
Problem 3: Evaluate the indefinite integral
Z
sin(ln(x))
dx
x
Problem 4: Evaluate the definite integral
Z
1
x − x2 dx
0
Problem 5: Find the area of the region enclosed by the curves y = x and
y = x2 from x = 0 to x = 1.
Summer Term I
Kostadinov
MA124 Calculus II
Boston University
Problem 6: Evaluate the definite integral using the tables. Indicate the
number of the formula used, as well as the numeric values of the parameters:
Z
π/6
e2x cos 3x dx
0
Problem7: Evaluate the definite integral using the tables. Indicate the number of the formula used, as well as the numeric values of the parameters:
Z
π/3
sin(2y) cos(y) dy
π/6
Problem8: Evaluate the definite integral using the tables. Indicate the number of the formula used, as well as the numeric values of the parameters:
Z
u2 + 3
√
du =
u2 u 2 + 1
Problem
9: Evaluate the improper integral
R∞ 1
dx
1 x2
Problem 10: Use Fundamental theorem of Calculus to find the derivative of
the function:
Rx
a) F (x) = 0 t2 dt
R x2
b) F (x) = 0 tan(t)dt
Summer Term I
Kostadinov
MA124 Calculus II
Boston University
TABLE OF INTEGRALS
√
√
R√
2
1:
a2 + u2 du = u2 a2 + u2 + a2 ln(u + a2 + u2 ) + C
√
R du
√
2:
=
ln(u
+
a2 + u2 ) + C
u2 +a2
R
2
2
√du
= − a a+u
3:
+C
2u
u2 u2 +a2
√
√
R √a2 +u2
a2 +u2
4:
du
=
−
+
ln(u
+
a2 + u 2 ) + C
2
u
u
√
R du
√
5:
= ln(u + a2 + u2 ) + C
a2 +u2
√
√
R u2 du
2
√
= − u2 u2 − a2 + a2 ln |u + u2 − a2 | + C
6:
u2 −a2
√
R u2 du
2
4
2
2
√
7:
=
(8a
+
3u
+
4a
u)
u − a2 + C
15
u−a2
R
8:
sin(au) cos(bu) du = − cos((a−b)u)
− cos((a+b)u)
+C
2(a−b)
2(a+b)
√
√
R u2 du
2
√
9:
= u2 a2 + u2 − a2 ln(u + a2 + u2 ) + C
a2 +u2
R
10:
u cos(u) du = cos(u) + u sin(u) + C
R au
au
11:
e cos bu du = a2e+b2 (a cos(bu) + b sin(bu)) + C
R au
12:
ue du = a12 (au − 1)eau + C
√
R
a+ a2 +u2
1
√ du
13:
ln(
)+C
=
−
a
u
u a2 +u2
Values of trigonometric functions, you may find useful:
sin(0) = 0
√
π
3
sin =
3
2
sin
π
=1
2
√
π
3
cos =
6
2
sin π = 0
cos
π
1
=
3
2