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Pattern - 21 of Indefinite Integration

February 19, 2021
Category: Pattern - 21 of Indefinite Integration

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⋆ Pattern – 21

$\int\frac{1}{\left(ax+b\right)^m\ \left(cx+d\right)^n}dx;$ where, m+n=2

1) I = $\int\frac{1}{\left(2x+3\right)^\frac{5}{7}\ \left(7x-5\right)^\frac{9}{7}}\ dx$

= $\int\frac{1}{2^\frac{5}{7}\ \left(x+\frac{3}{2}\right)^\frac{5}{7}\times7^\frac{9}{7}\ \left(x-\frac{5}{7}\right)^\frac{9}{7}}\ dx$

= $\frac{1}{2^\frac{5}{7}\times 7^\frac{9}{7}}\ \ \int\frac{1}{\left(x+\frac{3}{2}\right)^\frac{5}{7}\ \left(x-\frac{5}{7}\right)^\frac{9}{7}}\ dx$

= $\frac{1}{2^\frac{5}{7}\times 7^\frac{9}{7}}\ \int\frac{1}{\left(x+\frac{3}{2}\right)^{\frac{5}{7}+\frac{9}{7}}\ \left(\frac{x-\frac{5}{7}}{x+\frac{3}{2}}\right)^\frac{9}{7}}$

= $\frac{1}{2^\frac{5}{7}\times 7^\frac{9}{7}}\ \int\frac{1}{\left(x+\frac{3}{2}\right)^2\ \left(\frac{x-\frac{5}{7}}{x+\frac{3}{2}}\right)^\frac{9}{7}}\ dx$

Let, $\frac{x-\frac{5}{7}}{x+\frac{3}{2}}=y$

$\therefore\ \ \left[\frac{\left(x+\frac{3}{2}\right)\left(1\right)-\left(x-\frac{5}{7}\right)\left(1\right)}{\left(x+\frac{3}{2}\right)^2}\right]\ dx=dy$

$\therefore\ \ \frac{x+\frac{3}{2}-x+\frac{5}{7}}{\left(x+\frac{3}{2}\right)^2}\ dx=dy$

$\therefore\ \ \frac{31}{14\left(x+\frac{3}{2}\right)^2}\ dx=dy$

$\therefore\ \ \frac{1}{\left(x+\frac{3}{2}\right)^2}\ dx=\frac{14}{31}dy$

$\therefore\ \ I=\ \frac{14}{31\times 2^\frac{5}{7}\times 7^\frac{9}{7}}\ \int\frac{1}{y^\frac{9}{7}}\ dy$

$\therefore\ \ I=\ \frac{14}{31\times 2^\frac{5}{7}\times7^\frac{9}{7}}\ \int{y^\frac{-9}{7}\ }dy$

$\therefore\ \ I=\ \frac{14}{31\times2^\frac{5}{7}\times7^\frac{9}{7}}\times\ \frac{y^\frac{-1}{7}}{\frac{-2}{7}}+c$

$\therefore\ \ I=\ \frac{-49}{31\times2^\frac{5}{7}\times7^\frac{9}{7}\times y^\frac{2}{7}}+c$

$\therefore\ \ I=\ \frac{-7^\frac{5}{7}}{31\times 2^\frac{5}{7}\times y^\frac{2}{7}}+c$

$\therefore\ \ I=\ -\left(\frac{7}{2}\right)^\frac{5}{7}\times \frac{1}{31\times y^\frac{2}{7}}+c$

$\therefore\ \ I=\ -\left(\frac{7^5}{25y^2}\right)^\frac{1}{7}\times \frac{1}{31}+c$

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Pattern - 21 of Indefinite Integration

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