Integration by Parts

If u and v be two functions of x, then integral of product of these two functions is given by:

$\displaystyle \int u v dx = u \int v dx – \int \{\frac{d u}{dx}\int v dx \} dx$

Note:

In applying the above rule care has to be taken in the selection of the first function(u) and the second function (v).

Normally we use the following methods:

(i) If in the product of the two functions, one of the functions is not directly integrable (e.g. lnx, sin-1x, cos-1x, tan-1x etc.) then we take it as the first function and the remaining function is taken as the second function.

e.g. In the integration of ∫x tan-1x dx, tan-1x is taken as the first function and x as the second function.

(ii) If there is no other function, then unity is taken as the second function

e.g. In the integration of (tan-1x dx, tan-1x is taken as the first function and 1 as the second function).

(iii) If both of the function are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually we use the following preference order for the first function

(Inverse, Logarithmic, Algebraic, Trigonometric, Exponential)

In the above stated order, the function on the left is always chosen as the first function. This rule is called as :  ILATE
e.g. In the integration of ∫xsinxdx , x is taken as the first function and sinx is taken as the second function.

Illustration : Evaluate

(i) $\displaystyle \int lnx dx$

(ii) $\displaystyle \int \sqrt{x^2 + a^2} dx$

(iii) $\displaystyle \int \sqrt{a^2 – x^2} dx$

Solution:

(i) $\displaystyle I = \int lnx dx$

$\displaystyle = lnx \int 1.dx – \int (\frac{d(lnx)}{dx} \int 1. dx )dx$

$\displaystyle = x lnx – x + c$

(ii) $\displaystyle I = \int \sqrt{x^2 + a^2} dx$

$\displaystyle I = \sqrt{x^2 + a^2}\int 1.dx – \int (\frac{d(\sqrt{x^2 + a^2})}{dx} \int 1. dx)dx$

$\displaystyle I = x \sqrt{x^2 + a^2} – \int \frac{2 x}{2\sqrt{x^2 + a^2}} x dx$

$\displaystyle I = x \sqrt{x^2 + a^2} – \int \frac{ x^2}{\sqrt{x^2 + a^2}} dx$

$\displaystyle I = x \sqrt{x^2 + a^2} – \int \frac{ x^2+a^2}{\sqrt{x^2 + a^2}} dx + \int \frac{ a^2}{\sqrt{x^2 + a^2}} dx$

$\displaystyle I = x \sqrt{x^2 + a^2} – \int \sqrt{x^2 + a^2} + \int \frac{ a^2}{\sqrt{x^2 + a^2}} dx$

$\displaystyle I = x \sqrt{x^2 + a^2} – I + \int \frac{ a^2}{\sqrt{x^2 + a^2}} dx$

$\displaystyle 2I = x \sqrt{x^2 + a^2} + \int \frac{ a^2}{\sqrt{x^2 + a^2}} dx$

$\displaystyle I = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} ln(x+\sqrt{x^2 + a^2}) + c$

(iii) $\displaystyle I = \int \sqrt{a^2 – x^2} dx$

Using Integration by Parts

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