Mathematics

Integrate $$\displaystyle \int \dfrac{2w+3}{3w+4}dw$$.


SOLUTION
$$ \displaystyle I = \int \frac{2w+3}{3w+4}dw = 2\int \frac{w}{3w+4}dw+3\int \frac{dw}{3w+4}$$

Let $$ 3w+4 = u \Rightarrow du = 3dw $$

$$ \displaystyle  I = 2\int \frac{(u-4)}{3u} \frac{du}{3}+3\int \dfrac 1 {3u} du+c$$

$$ \displaystyle = \frac{2}{9}  \left( u-4 log u \right )+\frac{3log \left | 3w+4 \right |}{3}+c $$

$$ \displaystyle = \frac{2}{9}(3w+4)-\frac{-8}{9}log\left | 3w+4 \right |+\frac{3 log\left | 3w+4 \right |}{3}+c$$

$$ \displaystyle I = \dfrac{2}{9}(3w+4)+\frac{1}{9}log \left | 3w+4 \right |+c $$
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