∫3x+91 xdx =∫3(1+x)1 xdx = 3∫( x 1)1 xdx = 3∫1 1 x 11 xdx = 3∫1 x1 xdx+3∫2 dx1 x = 3∫dx 6∫dxx 1 = 3x 6log(x 1)+c ...

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Implicit Differentiation

Integrate ∫3...

Question

Integrate
$\int \frac{3 x + 9}{1 - x} d x$

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Solution

$\int \frac{3 x + 9}{1 - x} d x$
$= \int \frac{3 (1 + x)}{1 - x} d x$
$= - 3 \int \frac{(- x - 1)}{1 - x} d x$
$= - 3 \int \frac{1 - 1 - x - 1}{1 - x} d x$
$= - 3 \int \frac{1 - x}{1 - x} d x + 3 \int \frac{2 d x}{1 - x}$
$= - 3 \int d x - 6 \int \frac{d x}{x - 1}$
$= - 3 x - 6 log (x - 1) + c$

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