First slide

Methods of integration

Question

If $f (x) = \sqrt{4 x^{2} + 4 x - 3}$ then $\int \frac{x + 3}{f (x)} d x$ is equal to

Moderate

A

$(1 / 4) f (x) + (2 / 3) \log | 2 x + 1 + f (x) | + C$
B

$(1 / 4) f (x) + (5 / 4) \log | 2 x + 1 + f (x) | + C$
C

$(1 / 3) f (x) + (2 / 3) \log | (x + 3) + f (x) | + C$
D

$(2 / 3) f (x) + (1 / 3) \log | (x + 3) + f (x) |$

Solution

Put $2 x + 1 = t,$ so that $f (x) = \sqrt{t^{2} - 4}$ then
$\begin{array}{l} I = \frac{1}{4} \int \frac{t + 5}{\sqrt{t^{2} - 4}} d t \\ = \frac{1}{4} \sqrt{t^{2} - 4} + \frac{5}{4} \log |t + \sqrt{t^{2} - 4}| + C \\ = \frac{1}{4} f (x) + \frac{5}{4} \log | 2 x + 1 + f (x) | + C . \end{array}$

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