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Find the value of \( \int_{-3}^{0} \frac{d x}{(x+7) \sqrt{x+5}} \) |
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Answer» \(\int\limits_{-3}^0\frac{dx}{(x+7)\sqrt{x+5}}\) Let x + 5 = t2 ∴ dx = 2tdt x + 7 = t2 + 2 Let I = \(\int\frac{dx}{(x+7)\sqrt{x+5}}\) Then I = \(\int\frac{2tdt}{t(t^2+2)}\) = 2\(\frac{dt}{t^2+2}\) = \(\sqrt2 tan^{-1}(\sqrt{\frac{x+5}{2}})\) ∴ \(\int\limits_{-3}^0\frac{dx}{(x+7)\sqrt{x+5}}\) = \(\left[\sqrt2tan^{-1}\sqrt{\frac{x+5}2}\right]_{-3}^0\) = √2 tan-1\(\sqrt{\frac52}-\sqrt2 tan^{-1}1\) = √2 tan-1\(\left(\cfrac{\sqrt{\frac52}-1}{1+\sqrt{\frac52}}\right)\) = √2 tan-1 \((\frac{\sqrt5-\sqrt2}{\sqrt5+\sqrt2})\) |
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