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Question

Mathematics Question on integral

Integrate the function: 1xax−x2\frac{1}{x\sqrt{ax-x^2}}

Answer

1xax−x2\frac{1}{x\sqrt{ax-x^2}}

Let x=at\frac{a}{t} ⇒\Rightarrow dx=−at2-\frac{a}{t^2}dt

⇒\Rightarrow∫1xax−x2dx\frac{1}{x\sqrt{ax-x^2}}dx=∫1ata.at−(at)2(−at2)dt\frac{1}{\frac{a}{t}\sqrt{\frac{a.a}{t}}-(\frac{a}{t})^2}(\frac{-a}{t^2})dt

=−1a1t2t−t2t=-\frac{1}{a}{\frac{1}{\sqrt{\frac{t^2}{t}-\frac{t^2}{t}}}}

=−1a∫1t−1dt=-\frac{1}{a}\int \frac{1}{\sqrt{t-1}}dt

=−1a[2t−1]+C=-\frac{1}{a}[2\sqrt t-1]+C

=−1a[2ax−1]+C=-\frac{1}{a}[\frac{2\sqrt a}{x-1}]+C

=−2a(a−xx)+C=-\frac{2}{a}(\frac{\sqrt{a-x}}{x})+C