MA 3972-MA-Book April 11, 2018 15:57Def inite Integrals 259Example 2Evaluate∫ 40∣∣
x^2 − 4∣∣
dx.Setx^2 − 4 =0;x=±2.Thus,∣∣
x^2 − 4∣∣
={
x^2 −4ifx≥2orx≤− 2
−(x^2 −4) if− 2 <x< 2.
Thus,∫ 40∣∣
x^2 − 4∣∣
dx=∫ 20−(x^2 −4)dx+∫ 42(x^2 −4)dx=
[
−x^3
3+ 4 x] 20+
[
x^3
3− 4 x] 42=(
− 23
3+4(2)
)
−(0)+(
43
3−4(4)
)−
(
23
3−4(2)
)=
(
− 8
3+ 8
)
+(
64
3− 16
)
−(
8
3− 8
)
= 16.Verify your result with a calculator.TIP • You are not required to clear the memories in your calculator for the exam.
Definite Integrals Involving Trigonometric, Logarithmic,
and Exponential FunctionsExample 1Evaluate∫π0(x+sinx)dx.Rewrite:∫π0(x+sinx)dx=
x^2
2−cosx]π0=
(
π^2
2−cosπ)
−( 0 −cos 0)=
π^2
2+ 1 + 1 =
π^2
2+ 2.
Verify your result with a calculator.
Example 2Evaluate∫π/ 2π/ 4csc^2 ( 3 t)dt.Letu= 3 t;du= 3 dtor
du
3=dt.