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[parent] a lecture on integration by parts (Feature)

"a lecture on integration by parts" is owned by alozano.
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See Also: a lecture on integration by substitution, A lecture on trigonometric integrals and trigonometric substitution, a lecture on the partial fraction decomposition method, example of integration by parts involving algebraic manipulation


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a special case of partial integration (Feature) by pahio
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Cross-references: right, solution, order, integration by parts, composition, integral, integrate, derivative, formula, substitution, functions, product, integrand, product rule, indefinite integrals
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This is version 3 of a lecture on integration by parts, born on 2006-01-26, modified 2007-12-07.
Object id is 7574, canonical name is ALectureOnIntegrationByParts.
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AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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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
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