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Value of \( \int_{1}^{5}(\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2(x-1)}) d x \) is (a) \( \frac{8}{3} \) (b) \( \frac{16}{3} \) (c) \( \frac{32}{3} \) (d) \( \frac{34}{3} \) |
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Answer» Let I = \(\int\limits_1^5(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}})dx\) Let x - 1 = t2 then dx = 2tdt and limit converts to t = 0 to t = 2 \(\therefore\) I = \(\int\limits_0^22t(\sqrt{t^2+1+2t}+\sqrt{t^2+1-2t})dt\) \(=\int\limits_0^22t(\sqrt{(t+1)^2}+\sqrt{(t-1)^2})dt\) \(=\int\limits_0^22t({(t+1)}+{(t-1)})dt\) \(=\int\limits_0^22t\times2tdt = \int\limits_0^24t^2dt\) = \(\frac43(t^2)_0^2=\frac43(263 - 0) = \frac{32}3\) |
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