Question

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Answer 1

Given Integrand:-

$\displaystyle \sf \int \dfrac{dx}{(x - 1) \sqrt{x + 2} }$

Let t² = x + 2.

Differentiating w.r.t x on both sides:-

$\longrightarrow$ 2t. dt = dx

Thus,

$\implies \ \displaystyle \sf \int \dfrac{2 \cancel{t} }{( {t}^{2} - 2 - 1) \sqrt{ \cancel{{t}^{2} }} } dt \\ \\ \implies \ \displaystyle \sf 2\int \dfrac{dt}{ {t}^{2} - 3} \\ \\ \implies \ \displaystyle \sf 2\int \dfrac{dt}{ {t}^{2} - ( \sqrt{3}) {}^{2} }$

Of the form,

$\star \ \boxed{ \boxed{ \sf \int \dfrac{dx}{ {x}^{2} - {a}^{2} } = \dfrac{1}{2a} log \bigg| \frac{x - a}{x + a} \bigg| }}$

Thus,

$\implies \sf \: \dfrac{1}{2 \sqrt{3} } log \bigg( \dfrac{x - \sqrt{3} }{x + \sqrt{3} } \bigg) + C$

Answer 2

☘️ See the attachment picture for Explanation.

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