Lecture 6: Techniques of Integration
III. Trig Substitution, part II: Intergrate
the more general square root expressions
√
ax2 + bx + c, b 6= 0
Complete the Square first:
Z
x
√
dx
ex. 1.
2
3 − 2x − x
√
(− 3 − 2x − x2 − arcsin( x+1
2 ) + c)
1
Z
ex. 2.
x2
√
dx
2
4x − x√
√
1
2
2
(6 arcsin( x−2
2 ) − 2 (x − 2) 4x − x − 4 4x − x + c)
2
ex. 3.
Z p
( 92 [arcsin( x−2
3 )+
x−2
3
√
5 + 4x − x2dx
5+4x−x2
]
3
+ c)
3
NYTI: Evaluate the following integrals
Z
1
√
•
dx (− + c)
2
2
x x +4
√
Z √
•
Z
•
9 − x2
dx
2
x
√
(−
x2 +4
4x
9−x2
x
− arcsin( x3 ) + c)
1
dx
(5 − 4x − x2)5/2
1 √ x+2
x+2
( 81
( 5−4x−x2 + 13 ( √5−4x−x
)3 ) + c)
2
Z √
•
1 + x2
dx
x
√
√
2
( x2 + 1 + ln | xx+1 − x1 | + c)
4