मान लीजिए x neq frac{-3}{5}, frac{2}{5} है। य | vedclass

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(C) दिया गया है $f\left(\frac{2x+1}{5x+3}\right) = x+2$। मान लीजिए $t = \frac{2x+1}{5x+3}$। तब $t(5x+3) = 2x+1 \implies 5xt + 3t = 2x+1 \implies x(5t-

Vedclass · Accepted Answer

$\frac{7}{5}x - \frac{1}{25} \log |5x-2| + c$

Vedclass · Answer

(C) दिया गया है $f\left(\frac{2x+1}{5x+3}\right) = x+2$। मान लीजिए $t = \frac{2x+1}{5x+3}$। तब $t(5x+3) = 2x+1 \implies 5xt + 3t = 2x+1 \implies x(5t-2) = 1-3t \implies x = \frac{1-3t}{5t-2} = \frac{3t-1}{2-5t}$। फलन में $x$ का मान रखने पर: $f(t) = \frac{3t-1}{2-5t} + 2 = \frac{3t-1 + 4-10t}{2-5t} = \frac{3-7t}{2-5t} = \frac{7t-3}{5t-2}$। अब, $\int f(x) dx = \int \frac{7x-3}{5x-2} dx$। विभाजन का उपयोग करने पर: $\frac{7x-3}{5x-2} = \frac{7}{5} \left(\frac{5x-2}{5x-2}\right) + \frac{\frac{14}{5}-3}{5x-2} = \frac{7}{5} - \frac{1}{5(5x-2)}$। अतः, $\int f(x) dx = \int \left(\frac{7}{5} - \frac{1}{5(5x-2)}\right) dx = \frac{7}{5}x - \frac{1}{25} \ln |5x-2| + c$।

मान लीजिए $x \neq \frac{-3}{5}, \frac{2}{5}$ है। यदि $f\left(\frac{2x+1}{5x+3}\right) = x+2$ है, तो $\int f(x) dx =$

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