Texts:
WHAT DO YOU CALL AN UNCONTROLLABLE URGE TO PERFORM DIFFERENTIATION AND INTEGRATION?
Integration By Parts:
∫udv - uv - ∫vdu
∫xsin(x)dx - ∫xIn(x)dx
= let u = x and du = dx
let dv = sin(x)dx and v = -cos(x)
∫xsin(x)dx - (-xcos(x) + ∫cos(x)dx)
= x∫sin(x)dx + xcos(x) - ∫cos(x)dx
= x(-cos(x)) + xcos(x) + sin(x) + C
= -xcos(x) + xcos(x) + sin(x) + C
= sin(x) + C
Match each integral with a choice of u and v
a) Match with ∫udv - uv - ∫vdu
[∫xcos(x)dx
b) Match with ∫udv - uv - ∫vdu
[2∫xdx
c) Match with ∫udv - uv - ∫vdu
[3∫xcos(x)dx
d) Match with ∫udv - uv - ∫vdu
[4∫xsin(x)dx
e) Match with ∫udv - uv - ∫vdu
[∫edx
f) Match with ∫udv - uv - ∫vdu
[∫In(x)dx
g) Match with ∫udv - uv - ∫vdu
[7∫In(x)dx
h) Match with ∫udv - uv - ∫vdu
[3∫sin^2(x)dx
Answers for part a:
A. U = x, dv = cos(x)dx
B. U = x, dv = cos(x)dx
C. U = x, dv = sin(x)dx
D. U = x, dv = sin(x)dx
Answers for part b:
E. U = x, dv = edx
F. U = x, dv = edx
G. U = 3x, dv = sin^2(x)dx
H. U = 2x, dv = dx
V = e^x
V = e^x
V = -cos(2x)
V = x
U = x^2, dv = In(x)dx
U = x^2, dv = In(x)dx
U = x, dv = sin(x)dx
