28 Feb 2018
School
Department
Course
Professor
Miami University
MTH 151
Calculus I
Fall 2017
Term Test 3
Prof: Olga Brezhneva
Exam Guide
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
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
x√x−1
2
x sec−1 +C
−1
x√x−1
2
x csc−1 +C
Table of Antiderivatives
●In which is a function and is the constant(x)f C
Derivative Rules
●(x)±g(x)(f(x)±g(x))′=f′′
oAddition and subtraction rule