35 lines
594 B
TeX
35 lines
594 B
TeX
|
|
\newpage
|
|
|
|
\noindent
|
|
Here is a useful trick.
|
|
Difficult integrals involving sine and cosine
|
|
can often be solved by using exponentials.
|
|
Trigonometric simplifications involving powers
|
|
and multiple angles turn into simple algebra in the
|
|
exponential domain.
|
|
For example, the definite integral
|
|
$$\int_0^{2\pi}\left(\sin^4t-2\cos^3(t/2)\sin t\right)dt$$
|
|
can be solved as follows.
|
|
|
|
\medskip
|
|
\verb$f=sin(t)^4-2*cos(t/2)^3*sin(t)$
|
|
|
|
\verb$f=circexp(f)$
|
|
|
|
\verb$defint(f,t,0,2*pi)$
|
|
|
|
$$-{16\over5}+{3\over4}\pi$$
|
|
|
|
\medskip
|
|
\noindent
|
|
Here is a check.
|
|
|
|
\medskip
|
|
\verb$g=integral(f,t)$
|
|
|
|
\verb$f-d(g,t)$
|
|
|
|
$$0$$
|
|
|