Integrate -2w-33w-4dw-

I = ∫2w+3/3w+4dw = 2∫w/3w+4dw+3∫dw/3w+4 Let 3w+4 = u ⇒ du = 3dw #160; I = 2∫(u 4)/3udu/3+3∫ 1 3u du+c = 2/9 #160; ( u 4 log u )+3log | 3 ...

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Standard XII

Mathematics

Integration by Partial Fractions

Integrate ∫...

Question

Integrate $\int \frac{2 w + 3}{3 w + 4} d w$ .

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Solution

$I = \int \frac{2 w + 3}{3 w + 4} d w = 2 \int \frac{w}{3 w + 4} d w + 3 \int \frac{d w}{3 w + 4}$

Let $3 w + 4 = u \Rightarrow d u = 3 d w$

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

$= \frac{2}{9} (u - 4 l o g u) + \frac{3 l o g | 3 w + 4 |}{3} + c$

$= \frac{2}{9} (3 w + 4) - \frac{- 8}{9} l o g | 3 w + 4 | + \frac{3 l o g | 3 w + 4 |}{3} + c$

$I = \frac{2}{9} (3 w + 4) + \frac{1}{9} l o g | 3 w + 4 | + c$

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Integrate ∫2w+33w+4dw.

I=∫2w+33w+4dw=2∫w3w+4dw+3∫dw3w+4Let 3w+4=u⇒du=3dwI=2∫(u−4)3udu3+3∫13udu+c=29(u−4logu)+3log|3w+4|3+c=29(3w+4)−−89log|3w+4|+3log|3w+4|3+cI=29(3w+4)+19log|3w+4|+c

Integrate $\int \frac{2 w + 3}{3 w + 4} d w$ .