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Pattern - 25 of Indefinite Integration

February 19, 2021
Category: Pattern - 25 of Indefinite Integration

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⋆ Pattern – 25

$\int\frac{ae^x+be^{-x}+n}{pe^x+qe^{-x}+r}\ dx$

where, a, b , n, p, q, r ∈ R then use

$Nr=A\left(Dr\right)+B\frac{d}{dx}\left(Dr\right)+c$

1. $\ Let,\ I=\ \int\frac{4e^x+7e^{-x}-2}{3e^x-5e^{-x}+5}\ dx$

Let,

$4e^x+7e^{-x}-2=A\ \left(3e^x-5e^{-x}+5\right)+B\left(3e^x+5e^{-x}\right)+c$

$\therefore\ \ 4e^x+7e^{-x}-2=\ \left(3A-3B\right)e^x+\left(-5A+5B\right)e^{-x}+\left(5A+c\right)$

∴ Comparing

3A + 3B = 4 —–(1)

-5A + 5B = 7 ——(2)

5A + c = -2 ——–(3)

From (1) & (2)

15A + 15B = 20

-15A + 15B =- 21

30B = 41

$\therefore\ \ B=\ \frac{41}{30}$

From (1)

$3A+\ \frac{123}{30}=4$

$\therefore\ \ 3A=4-\frac{123}{30}$

$\therefore\ \ 3A=\ \frac{-3}{30}$

$\therefore\ \ A=\ -\frac{1}{30}$

From (3),

$\frac{-5}{30}+c=\ -2$

$\therefore\ \ c=-2+\frac{5}{30}$

$=\ \frac{-55}{30}$

Hence,

$e^x+7e^{-x}-2=\ \frac{-1}{30}\left(3e^x-5e^{-x}+5\right)+\frac{41}{30}\left(3e^x+5e^{-x}\right)-\frac{55}{30}$

$\therefore\ \ I=\ \int\left(\frac{-1}{30}\right)dx+\frac{41}{30}\int\frac{3e^x+5e^{-x}}{3e^x-5e^{-x}+5}\ dx-\frac{55}{30}\int\frac{1}{3e^x-5e^{-x}+5\ }dx$

$=\ \frac{-1}{30\ }\int1dx+\frac{41}{30}\int\frac{3e^x+5e^{-x}}{3e^x-5e^{-x}+5}dx-\frac{55}{30}I_1----\left(4\right)$

Now,

$I_1=\ \int\frac{1}{3e^x-5e^{-x}+5}\ dx$

$=\ \int\frac{e^x}{3\left(e^x\right)^2-5+5e^x}\ dx$

( Let, $e^x$ = t ∴ $e^x$ dx = dt)

$=\ \int\frac{1}{3t^2+5t-5}dt$

$=\ \int\frac{1}{3t^2+5t+\frac{25}{12}-\frac{15}{12}-5}\ dt$

$=\int\frac{1}{\left(\sqrt3t+\frac{5}{2\sqrt3}\right)^2-\ \frac{85}{12}}\ dt$

$=\ \frac{1}{2\sqrt3\left(\frac{\sqrt{85}}{2\sqrt3}\right)}\times\frac{1}{\sqrt3}\times\log{\left|\frac{\sqrt3t+\ \frac{5}{2\sqrt3}-\ \frac{\sqrt{85}}{2\sqrt{3\ }}}{\sqrt3t+\ \frac{5}{2\sqrt3}+\ \frac{85}{2\sqrt3}}\right|}$

$=\ \frac{1}{\sqrt{85}}\log{\left|\frac{6t+5-\sqrt{85}}{6t+5+\sqrt{85}}\right|}$

$=\ \frac{1}{\sqrt{85}}\log{\left|\frac{6e^x+5-\sqrt{85}}{6e^x+5+\ \sqrt{85}}\right|}$

$=\frac{-1}{30}\ \int1dx+\frac{41}{30}\ \int\frac{3e^x+5e^{-x}}{3e^x-5e^{-x}+5}dx-\frac{55}{30}I_1$

& $\ \frac{1}{\sqrt{85}}\log{\left|\frac{5e^x+5-\sqrt{85}}{6c^x+5+\sqrt{85}}\right|}$

So,

$=\ \frac{-x}{30}+\frac{41}{30}\log{\left|3e^x-5e^{-x}+5\right|}-\frac{55}{20\sqrt{85}}\log{\left|\frac{6e^x+5-\sqrt{85\ }}{6e^x+5+\sqrt{85}}\right|}+c$