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Question
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
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Solution
\[Let I = \int_1^5 \frac{x}{\sqrt{2x - 1}} d x\]
\[Let, 2x - 1 = t,\text{ then }2dx = dt, \]
\[\text{When, }x \to 1 ; t \to 1\text{ and x to 5; } t \to 9\]
\[x = \frac{t + 1}{2}\]
\[I = \frac{1}{2} \int_1^9 \frac{t + 1}{\sqrt{t}} \times \frac{dt}{2}\]
\[ = \frac{1}{4} \left[ \frac{2 t^\frac{3}{2}}{3} + 2\sqrt{t} \right]_1^9 \]
\[ = \frac{1}{4}\left[ 18 + 6 - \frac{2}{3} - 2 \right]\]
\[ = \frac{16}{3}\]
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