First slide

Introduction to integration

Question

If $f (x) = \sqrt{x^{2} - a^{2}}$ then $\int \frac{x^{2}}{f (x)} d x$ is

Easy

A

$x f (x) + \frac{a^{2}}{2} \log | x + f (x) | + C$
B

$\frac{x^{2}}{2} f (x) - \frac{a^{2}}{2} \log | x + f (x) | + C$
C

$\frac{x^{2}}{2} f (x) + a^{2} \log | x - f (x) | + C$
D

$\frac{x}{2} f (x) + \frac{a^{2}}{2} \log | x + f (x) | + C$

Solution

$\begin{array}{l} I = \int \frac{x^{2}}{\sqrt{x^{2} - a^{2}}} d x \\ = \int \sqrt{x^{2} - a^{2}} d x + a^{2} \int \frac{d x}{\sqrt{x^{2} - a^{2}}} \\ = \frac{1}{2} x \sqrt{x^{2} - a^{2}} - \frac{1}{2} a^{2} \log |x + \sqrt{x^{2} - a^{2}}| \\ + a^{2} \log |x + \sqrt{x^{2} - a^{2}}| + c \\ = \frac{1}{2} x f (x) + \frac{1}{2} a^{2} \log | x + f (x) | + C \end{array}$

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