Special Tricks &Pitfalls of Integration
MATH 1920
Integration using the Evaluation Theorem is, in general, not easy. The problem is that for
most integrals, it is difficult to find the antiderivative of the integrand. We have looked at one
very useful technique (trick) to help: the substitution method. This process involves “undoing”
the Chain Rule. We will look at more techniques to help us when integrating in the coming
days.
Even if we cannot find an antiderivative, there are times when we can easily evaluate an
integral whose integrand is an EVEN or ODD function.
Suppose fis continuous on −a,a.Iffis even then f−xfxand
−a
afxdx 20
afxdx
If fis odd then f−x−fxand
−a
afxdx 0
Examples of even functions:
axnwhere nis even, cosx,secx
Examples of odd functions
axnwhere nis odd, sinx,cscx,tanx,cotx
Evaluate (see my notes for Section 5.5 for solutions):
1.−5
5x6dx
2.−10
10 y13 dy
3.−/2
/2 sin3xcos xsin xcos xdx
Beware of the pitfalls in integration. Explain what is wrong in each of the following
situations.
1.1
2x−3dx −3x−4|1
2−3
x41
2−3
243
1445
16
2.−2
11
xln|x||−2
1ln|1|−ln|−2|0−ln2 −ln2 ln 1
2
3.xcosxdx x2
2sinxC1
2x2sinxC
Please take note of the VARIABLE of integration. Evaluate the following.
1.r2hdh
2.r2hdr
3.35dx
4.e4dx
