MATH 1225 Study Guide - Spring 2018, Comprehensive Midterm Notes - Integral, Graph Of A Function, Trigonometric Functions
12 Oct 2018
School
Department
Course
Professor
MATH 1225
MIDTERM EXAM
STUDY GUIDE
Fall 2018
Sec. 5.5 The Substitution Rule
• The Substitution Rule: If
xgu
is a differentiable function whose range is an interval
I
and
f
is
continuous on
I
, then
•duufdxxgxgf
[Note: We can think of
dx
and
du
after integral signs as if they were differentials]
• The Substitution Rule for Definite Integrals: If
g
is continuous on
ba,
and
f
is continuous on the range
of
xgu
, then
•
b
a
bg
ag duufdxxgxgf
EX: Evaluate the following integral
1.
2.
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3.
4.
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Document Summary
Xg u : the substitution rule: if is a differentiable function whose range is an interval i and f is xgf xg continuous on i , then duuf dx. [note: we can think of dx and du after integral signs as if they were differentials: the substitution rule for definite integrals: if g is continuous on . Ba, and f is continuous on the range u of. , then xgf xg dx bg ag uf du b a. Ex: evaluate the following integral: +(cid:883, +(cid:883) (cid:2869)(cid:2868, (cid:2871) (cid:2870) (cid:883, (cid:2871) (cid:2870) (cid:883) (cid:2870)(cid:2869, ln(cid:4666)(cid:3119)(cid:4667)+(cid:2869) (cid:2871) x. Ex: evaluate the following integral: (cid:3119)(cid:4666)(cid:3118) (cid:2870)+(cid:2869)(cid:4667) (cid:2870)(cid:2869, (cid:884)(cid:2871)(cid:4666)(cid:2872)+(cid:883)(cid:4667)(cid:2872) (cid:2868) (cid:2869, (cid:2871) (cid:2869) (cid:3118) dx x. Integral of symmetric functions: suppose f is continuous on . If f is odd f x x f. 5. 4 continue indefinite integrals and the net change theorem: net change theorem: the integral of a rate of change is the net change: xf dx.
