Evaluate- - 1 x - a x - x 2

∫1x√(ax x^2)dx=∫dxx^2√(a/x 1) Put ax 1=t^2 ax^2dx=2tdt I=∫ 2ta(1t)dt I= 2at+c I= 2a√(ax 1)+c ...

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Standard IX

Mathematics

Multiplication of Surds

evaluate: ∫ ...

Question

evaluate:
$\int \frac{1}{x \sqrt{a x - x^{2}}}$

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Solution

$\int \frac{1}{x \sqrt{a x - x^{2}}} d x = \int \frac{d x}{x^{2} \sqrt{a / x - 1}}$
Put $\frac{a}{x} - 1 = t^{2} - \frac{a}{x^{2}} d x = 2 t d t$
$I = \int - \frac{2 t}{a} (\frac{1}{t}) d t I = - \frac{- 2}{a} t + c I = - \frac{2}{a} \sqrt{\frac{a}{x} - 1} + c$

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evaluate:∫1x√ax−x2

∫1x√ax−x2dx=∫dxx2√a/x−1Put ax−1=t2−ax2dx=2tdtI=∫−2ta(1t)dtI=−−2at+cI=−2a√ax−1+c

evaluate:
$\int \frac{1}{x \sqrt{a x - x^{2}}}$

$\int \frac{1}{x \sqrt{a x - x^{2}}} d x = \int \frac{d x}{x^{2} \sqrt{a / x - 1}}$
Put $\frac{a}{x} - 1 = t^{2} - \frac{a}{x^{2}} d x = 2 t d t$
$I = \int - \frac{2 t}{a} (\frac{1}{t}) d t I = - \frac{- 2}{a} t + c I = - \frac{2}{a} \sqrt{\frac{a}{x} - 1} + c$