DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COLLEGE OF SCIENCE
UNIVERSITY OF THE PHILIPPINES BAGUIO
MATHEMATICS 54 HOMEWORK
1st Semester AY 2016-2017
”Cheating is a form of intellectual dishonesty that will not be condoned. Anyone caught cheating shall be dealt with in accordance with the rules on
student conduct and discipline. Cheating is punishable with a grade of 5.0 in the course and/or (1) suspension for a period of not less than one year;
or (2) expulsion from the University.”
Z
4x2 − 3x + 2
dx =
4x2 − 4x + 3
Z x−1
1+ 2
dx
by long division
4x − 3x + 2
Z
x−1
= x + c1 +
dx
2
(4x − 4x + 1) + 2
Z
Z
dx
x
dx −
= x + c1 +
2
(2x − 1) + 2
(2x − 1)2 + 2
Let
u = 2x − 1
⇒ x = u+1
2
⇒ du = 2dx
Then,
Z
Z u+1
Z
4x2 − 3x + 2
1
1
du
2
dx = x + c1 +
du −
2
2
2
4x − 4x + 3
2
u +2
2
u +2
Z
Z
1
u+1
1
du
= x + c1 +
du −
2
2
4
u +2
2
u +2
Z
Z
Z
1
u
1
du
1
du
= x + c1 +
du +
−
2
2
2
4
u +2
4
u +2 2
u +2
Z
Z
1
u
1
du
= x + c1 +
du −
2
2
4
u +2
4
u +2
2x − 1
1
1
+c
= x + ln |4x2 − 4x + 1| − √ tan−1 √
8
4 2
2
√
√
1
2
2(2x − 1)
= x + ln |4x2 − 4x + 1| −
tan−1
+c
8
8
2
o.cawiding