|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
|
|
|
| ``Integration by parts and by substitution''
by perucho on 2006-01-28 14:31:27 |
|
| Complementing the interesting entries owned by Alozano, I must say that the substitution u=tan(x/2)is important in the integration of tigonometrical rational forms, for example, $\int\frac{1}{a+\cos{x}}dx$.
Sometimes is a good idea passing to z-plane. For instance, the integrals $\int e^{ax}\cos{bx}dx$ and
$\int_0^t(\alphat+v_0)\sin[\frac{\beta}{\alpha}\ln{\frac{\alpha}{v_0}t+1}]dt$ (notation's abuse is irrelevant here).
The former one is classic, but the latter arose from a nice problem of kinematics of a particle. It is easy to show that the last integral can be transformed into the inmediate integral
$\frac{1}{v_0^{\imath\frac{\beta}{\alpha}}}\int_0^t(\alphat+v_0)^{\imath\frac{\beta}{\alpha}+1}dt$.
(I used the formulas e^{\imathu}=\cosu+\imath\sinu (Euler) and
A^{\imathB}=\cos{B\ln{A}}+\imath\sin{B\ln{A}} to separate the real and imaginary parts).
On the order hand, integration by parts is often a suitable method in integrals of trigonometrical functions like, for example, $\int\sec^3{x}dx$, which is classic too.
Well,I think that some examples exposed by Alvaro are quite casuals but the idea is to justify the methods.
perucho |
| | [ reply | up ] |
|
|
|
|
|
|
