Question

\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]

Answer 1

\[Let I = \int_1^5 \frac{x}{\sqrt{2x - 1}} d x\]

\[Let, 2x - 1 = t,\text{ then }2dx = dt, \]

\[\text{When, }x \to 1 ; t \to 1\text{ and x to 5; } t \to 9\]

\[x = \frac{t + 1}{2}\]

\[I = \frac{1}{2} \int_1^9 \frac{t + 1}{\sqrt{t}} \times \frac{dt}{2}\]

\[ = \frac{1}{4} \left[ \frac{2 t^\frac{3}{2}}{3} + 2\sqrt{t} \right]_1^9 \]

\[ = \frac{1}{4}\left[ 18 + 6 - \frac{2}{3} - 2 \right]\]

\[ = \frac{16}{3}\]