# MTH 151 Midterm: Term Test 3 - Fall 2017

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School
Department
Course
Miami University
MTH 151
Calculus I
Fall 2017
Term Test 3
Prof: Olga Brezhneva
Exam Guide
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Antiderivatives
Definition of Antiderivative
is an antiderivative of a function on interval F f I
If you differentiate , you get the original function F f
(x) (x)F=f
Theorem
If   is the antiderivative of on , then the most general antiderivative of on   isF f I f I
where   is an arbitrary constant(x)F+C C
All possible variations of constant C
Exercise
Find the most general antiderivative of
oA) in (x) s
os (x) c+C
oB) x
1
n|x|l+C
oC) xn
x⁄(n))( n+1 + 1 + C
oD) 3
1+x2
tan (x) 3 −1 +C
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Table of Antiderivatives
(x)f
(x)f
xn
n |x| l+C
k
xk +C
x
x+C
bx
bx
Inb
os (x) c
in x s+C
in (x) s
os x c+C
x sec2
an x t+C
x cos2
ot x c+C
ec xtan x s
ec x s+C
sc xcot x c
sc x c+C
1
1−x2
x sin−1 +C
−1
1−x2
x cos−1 +C
1
1+x2
x tan−1 +C
1
xx−1
2
x sec−1 +C
−1
xx−1
2
x csc−1 +C
Table of Antiderivatives
In which   is a function and   is the constant(x)f C
Derivative Rules
(xg(x)(f(xg(x))=f
oAddition and subtraction rule
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