If $$f(x)=x+22x+3$$. then ∫$$f(x)x21/2dxis$$ equal to $$12g1+2f(x)1$$−$$2f(x)$$−$$23h3f(x)+23f(x)$$−$$2$$ $$+C$$ where

Correct option is 4g(x)=log⁡|x|,h(x)=log⁡|x|

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If $f (x) = \frac{x + 2}{2 x + 3} .$ then $\int {(\frac{f (x)}{x^{2}})}^{1 / 2} d x$
is equal to $\frac{1}{\sqrt{2}} g (\frac{1 + \sqrt{2 f (x)}}{1 - \sqrt{2 f (x)}}) - \sqrt{\frac{2}{3}} h (\frac{\sqrt{3 f (x)} + \sqrt{2}}{\sqrt{3 f (x)} - \sqrt{2}})$
$+ C$ where

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g(x)=tan−1⁡x,h(x)=log⁡|x|

g(x)=log⁡|x|,h(x)=tan−1⁡x

g(x)=h(x)=tan−1⁡x

g(x)=log⁡|x|,h(x)=log⁡|x|

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detailed solution

Correct option is D

Putting y2=f(x)=x+22x+3, we have x=3y2−21−2y2 and dx=−2y1−2y22dy.So =−∫y⋅2y1−2y22⋅1−2y23y2−2dy=2∫y22y2−13y2−2dy=−2∫12y2−1−23y2−2dy=12log⁡1+2y1−2y−23log⁡3y+23y−2+CThus g(x)=log⁡|x| and h(x)=log⁡|x|.

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If f(x)=x+22x+3. then ∫f(x)x21/2dxis equal to 12g1+2f(x)1−2f(x)−23h3f(x)+23f(x)−2 +C where

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If $f (x) = \frac{x + 2}{2 x + 3} .$ then $\int {(\frac{f (x)}{x^{2}})}^{1 / 2} d x$
is equal to $\frac{1}{\sqrt{2}} g (\frac{1 + \sqrt{2 f (x)}}{1 - \sqrt{2 f (x)}}) - \sqrt{\frac{2}{3}} h (\frac{\sqrt{3 f (x)} + \sqrt{2}}{\sqrt{3 f (x)} - \sqrt{2}})$
$+ C$ where