∫dx(x+5)x+4 on (−4,∞)

Put x+4=t2⇒dx=2t dt Now x+5=t2+1 ∫2t(t2+1).tdt=2Tan−1(t)+C=2Tan−1(x+4)+C

∫dx(x+5)x+4 on (−4,∞)

Put x+4=t2⇒dx=2t dt Now x+5=t2+1 ∫2t(t2+1).tdt=2Tan−1(t)+C=2Tan−1(x+4)+C

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$\int \frac{d x}{(x + 5) \sqrt{x + 4}}$ on $(- 4, \infty)$

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$2 \cot^{- 1} (\sqrt{x + 4}) + C$

$T a n^{- 1} \sqrt{x + 5} + C$

$2 T a n^{- 1} \sqrt{x + 4} + C$

$2 \sin^{- 1} (\sqrt{x + 4}) + C$

answer is C.

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Detailed Solution

Put $x + 4 = t^{2} \Rightarrow d x = 2 t d t$
Now $x + 5 = t^{2} + 1$
$\int \frac{2 t}{(t^{2} + 1) . t} d t = 2 T a n^{- 1} (t) + C = 2 T a n^{- 1} (\sqrt{x + 4}) + C$

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$\int \frac{d x}{(x + 5) \sqrt{x + 4}}$ on $(- 4, \infty)$