First slide

Introduction to integration

Question

$Evaluate \int \frac{d x}{(2 x - 7) \sqrt{(x - 3) (x - 4)}}$

Easy

A

$\sec^{- 1} (2 x - 7) + c$
B

$\frac{1}{2} \sec^{- 1} (2 x - 7) + c$
C

$\frac{1}{7} \sec^{- 1} (2 x - 7) + c$
D

None of these

Solution

$\begin{array}{l} I = \int \frac{d x}{(2 x - 7) \sqrt{(x - 3) (x - 4)}} \\ = \int \frac{d x}{(2 x - 7) \sqrt{x^{2} - 7 x + 12}} \\ = \frac{1}{2} \int \frac{2 d x}{(2 x - 7) \sqrt{(2 x - 7)^{2} - 1}} \\ = \frac{1}{2} \sec^{- 1} (2 x - 7) + c (∵ \int \frac{d t}{|t| \sqrt{t^{2} - 1}} = s e c^{- 1} (t) + c) \end{array}$

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