Let I = ∫ #10;x^2( a + bx)^2dx #10; #10;Put a + bx = u , then bdx #10;= du and x = u ab . #10; #10; I = ∫ #10;( u ab ...

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Integration by Substitution

Solve ∫x2 a...

Question

Solve $\int \frac{x^{2}}{{(a + b x)}^{2}} d x$

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Solution

Let $I = \int \frac{x^{2}}{{(a + b x)}^{2}} d x$

Put $a + b x = u$ , then $b d x = d u$ and $x = \frac{u - a}{b}$ .

$I = \int \frac{{(\frac{u - a}{b})}^{2}}{{(u)}^{2}} \frac{d u}{b}$

$= \int \frac{u^{2} + a^{2} - 2 a u}{b^{3} u^{2}} d u$

$= \frac{1}{b^{3}} \int \frac{u^{2} + a^{2} - 2 a u}{u^{2}} d u$

$= \frac{1}{b^{3}} [\int d u + a^{2} \int \frac{1}{u^{2}} d u - 2 a \int \frac{1}{u} d u]$

$= \frac{1}{b^{3}} [u - \frac{a^{2}}{u} - 2 a log u] + c$

$= \frac{1}{b^{3}} [(a + b x) - \frac{a^{2}}{(a + b x)} - 2 a log | a + b x |] + c$

$= \frac{(a + b x)}{b^{3}} - \frac{a^{2}}{b^{3} (a + b x)} - \frac{2 a}{b^{3}} log | a + b x | + c$

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