If gleft(frac{t+1}{2 t+1}right)=t+1,then int | vedclass

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(D) Given $g\left(\frac{t+1}{2 t+1}\right)=t+1$. Let $x = \frac{t+1}{2 t+1}$. Then $x(2t+1) = t+1 \Rightarrow 2tx + x = t+1 \Rightarrow t(2x-1) = 1-x$

Vedclass · Accepted Answer

$\frac{x}{2}+\frac{1}{4} \log _e|2 x-1|+c$

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(D) Given $g\left(\frac{t+1}{2 t+1}\right)=t+1$. Let $x = \frac{t+1}{2 t+1}$. Then $x(2t+1) = t+1 \Rightarrow 2tx + x = t+1 \Rightarrow t(2x-1) = 1-x$. So,$t = \frac{1-x}{2x-1}$. Substituting $t$ into $g(x) = t+1$: $g(x) = \frac{1-x}{2x-1} + 1 = \frac{1-x+2x-1}{2x-1} = \frac{x}{2x-1}$. Now,$\int g(x) dx = \int \frac{x}{2x-1} dx$. $= \frac{1}{2} \int \frac{2x}{2x-1} dx = \frac{1}{2} \int \frac{2x-1+1}{2x-1} dx$. $= \frac{1}{2} \int \left(1 + \frac{1}{2x-1}\right) dx$. $= \frac{1}{2} \left(x + \frac{1}{2} \log _e|2x-1|\right) + C$. $= \frac{x}{2} + \frac{1}{4} \log _e|2x-1| + C$.

If $g\left(\frac{t+1}{2 t+1}\right)=t+1$,then $\int g(x) d x=$

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