236 STEP 4. Review the Knowledge You Need to Score High
3.
∫a
a
f(x)dx= 0
4.
∫b
a
f(x)dx=−
∫a
b
f(x)
5.
∫b
a
Cf(x)dx=C
∫b
a
f(x)dxwhenCis a constant.
6.
∫b
a
[f(x)±g(x)]dx=
∫b
a
f(x)dx±
∫b
a
g(x)dx
7.
∫b
a
f(x)dx≥0 provided f(x)≥0on[a,b].
8.
∫b
a
f(x)dx≥
∫b
a
g(x)dxprovided f(x)≥g(x)on[a,b].
9.
∣
∣∣
∣
∫b
a
f(x)dx
∣
∣∣
∣≤
∫b
a
∣∣
f(x)
∣∣
dx
10.
∫b
a
g(x)dx≤
∫b
a
f(x)dx≤
∫b
a
h(x)dx; providedg(x)≤f(x)≤h(x)on[a,b].
- m(b−a)≤
∫b
a
f(x)dx≤M(b−a); providedm≤ f(x)≤Mon [a,b].
12.
∫c
a
f(x)dx=
∫b
a
f(x)dx+
∫c
b
f(x)dx; providedf(x) is integrable on an interval
containinga,b,c.
Examples
1.
∫π
π
cosxdx= 0
2.
∫ 5
1
x^4 dx=−
∫ 1
5
x^4 dx
3.
∫ 7
− 2
5 x^2 dx= 5
∫ 7
− 2
x^2 dx
4.
∫ 4
0
(
x^3 − 2 x+ 1
)
dx=
∫ 4
0
x^3 dx− 2
∫ 4
0
xdx+
∫ 4
0
1 dx
5.
∫ 5
1
√
xdx=
∫ 3
1
√
xdx+
∫ 5
3
√
xdx
Note: Or
∫ 3
1
√
xdx=
∫ 5
1
√
xdx+
∫ 3
5
√
xdx
∫c
a
=
∫b
a
+
∫c
b
a, b, cdo not have to be arranged from smallest to largest.