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[parent] a special case of partial integration (Feature)

In determining the antiderivative of a transcendental function $ U$ whose derivative $ U'$ is algebraic, the result can be obtained when choosing in the formula

$\displaystyle \int UV'\,dx = UV-\!\int VU'\,dx$
of integration by parts $ V' \equiv 1$; then one has
$\displaystyle \int U\,dx = \int U\!\cdot\!1\,dx\; =\; U\!\cdot\!x-\!\int x\!\cdot\!U'\,dx.$
The functions $ U$ in question are mainly the logarithm, the cyclometric functions and the area functions.

Examples.

  1. $ \displaystyle\int\!\ln{x}\,dx = x\ln{x}-\!\int x\!\cdot\!\frac{1}{x}\,dx =\; x\ln{x}-x+C$
  2. $ \displaystyle\int\!\arcsin{x}\,dx = x\arcsin{x}-\!\int\!x\!\cdot\!\frac{1}{\sq... ...{1}{2}\int\!\frac{-2x}{\sqrt{1\!-\!x^2}}\,dx\\ = x\arcsin{x}+\sqrt{1\!-\!x^2}+C$
  3. $ \displaystyle\int\!\arctan{x}\,dx = x\arctan{x}-\!\int\!x\!\cdot\!\frac{1}{1\!... ...= x\arctan{x}-\frac{1}{2}\ln(1\!+\!x^2)+C =\; x\arctan{x}-\ln\sqrt{1\!+\!x^2}+C$
  4. $ \displaystyle\int\!{\mathrm{arcosh}}{x}\,dx = x{\mathrm{arcosh}}{x}-\!\int\!x\... ...t\!\frac{1}{\sqrt{x^2\!-\!1}}\,dx \\ = x{\mathrm{arcosh}}{x}-\sqrt{x^2\!-\!1}+C$

The choice $ V' \equiv 1$ works as well in such cases as $ \int(\ln{x})^2\,dx$ and $ \int\ln(\ln{x})\,dx$, giving respectively $ x((\ln{x})^2-2\ln{x}+2)+C$ and $ x\ln(\ln{x})-{\mathrm{Li}}{x}+C$ (see logarithmic integral). Also $ \int(\arcsin{x})^2\,dx$ succeeds, requiring two integrations by parts, and giving the result $ x(\arcsin{x})^2+2\sqrt{1\!-\!x^2}\arcsin{x}-2x+C$.



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"a special case of partial integration" is owned by pahio.
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See Also: table of integrals

Keywords:  integration by parts

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integration of $\sqrt{x^2+1}$ (Derivation) by pahio
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Cross-references: logarithmic integral, area functions, cyclometric functions, integration by parts, derivative, function, antiderivative
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This is version 8 of a special case of partial integration, born on 2007-11-27, modified 2008-02-22.
Object id is 10065, canonical name is ASpecialCaseOfPartialIntegration.
Accessed 572 times total.

Classification:
AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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