Integration by parts
Integration by parts is a mathematical technique used in the solving of indefinite and definite integrals. The differential functions of the products are expanded and then expressed in the original integrals (Stroud & Dexter, 70).
The single integration by parts begins with the following:
(uv)' = uv' + u'v and integration of both sides of the equation yields;
uv = ∫uv' dx + ∫u'v dx
Rearranging the integrals gives;
∫uv' dx = uv − ∫u'v dx
The key concepts required for easy computation of the integration by parts include;
The choice of dx and dx is such a way that;
u is easy to differentiate and dx is easy to integrate.
In addition, ∫Vdu and ∫udv should be easy to compute. Sometimes in integration by parts, it is necessary to integrate more than once.
Basic example of the integral by parts
Integrate ∫xcos(x) dx using the integration by parts
The first step is choosing properly the functions u and v
Therefore u =x and v= cos (x). The integral is now in the form ∫uv dx
Differentiating u = u’=x’ and integrating v= ∫v dx =∫ cos(x) dx= sin(x) as per the integration rules. The constant C during interaction by parts is always taken as 0 for purposes of convenience when determining a particular function. At times, there is a challenge in choosing u and v.
Example
Integrate ∫ln(x) dx using integration by parts.
There is only one function so it is very difficult in choosing u and v but how?
The solution is choosing v as 1 hence
u= ln(x); differentiating ln(x) gives 1/x and integrating v∫1dx yields x. Putting them together and simplifying gives; xln(x) –x +C.
Works Cited
Stroud, Kenneth Arthur, and Dexter J. Booth. Engineering mathematics. palgrave macmillan, 2013.
