First slide

Evaluation of definite integrals

Question

Value of $\int_{1}^{5} {\sqrt{x + 2 \sqrt{x - 1}} + \sqrt{x - 2 (x - 1)}} dx$ is

Difficult

A

$\frac{8}{3}$
B

$\frac{16}{3}$
C

$\frac{32}{3}$
D

$\frac{34}{3}$

Solution

$\begin{array}{l} \sqrt{x + (2 \sqrt{x - 1})} = | \sqrt{x - 1} + 1 | \\ \Rightarrow \sqrt{x - 2 \sqrt{x - 1}} = | \sqrt{x - 1} - 1 | \\ I = \int_{1}^{2} 2 dx + \int_{2}^{5} 2 \sqrt{x - 1} dx \\ = 2 \cdot 1 + 2 \cdot {[(x - 1)^{3 / 2} \cdot \frac{2}{3}]}_{2}^{5} \\ = 2 + \frac{28}{3} = \frac{34}{3} \end{array}$

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