MA 16500 Lecture Notes - Lecture 32: Antiderivative
6 Jun 2018
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What is an indefinite integral? It’s an integral expressed without limits while containing an
arbitrary constant
I.e. where (x) dx (x)
∫
f=F+C(x) (x)F′=F
0x sec (x) dx x tan(x) C
∫
14+ 2 2= 2 5+ 2 +
Ex: dt dt t t
∫
t2
2t+t−1
2 2√t=∫
2 + √t−t−2 = 2 + 3
22
3−t−1 +C
Ex: x dx
∫
2√x2+ 1
u=x2+ 1
u x dx >dx d= 2 = = 2x
du
=> du (x)
∫
√u=2
3u2
3+C=2
3 2 + 1 2
3+C
Ex: cos(x)dx
∫
x3 4 + 2
u=x4+ 2
u x dx >xd = 4 3=d=du
4x3
=> du
∫
4
cos(u)=4
sin(u)+C=4
sin(x+2)
4+C
Ex: (2x) dx
∫
+ 1 0.5
xu = 2 + 1
u dx >xd = 2 = d=2
du
u du (2x)
∫
2
1 0.5 =u
3
12
3+C=3
1+ 1 2
3+C
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