\[\int\sqrt{2ax - x^2} \text{ dx}\]

\[\text{ Let I }= \int\sqrt{2ax - x^2}\text{ dx}\]\[ = \int\sqrt{a^2 + 2ax - x^2 - a^2}\text{ dx}\]\[ = \int \sqrt{a^2 - \left( x^2 - 2ax + a^2 \right)}\text{ dx}\]\[ = \int\sqrt{a^2 - \left( x - a \right)^2}\text{ dx}\]\[ = \left( \frac{x - a}{2} \right) \sqrt{2ax - x^2} + \frac{a^2}{2} \sin^{- 1} \left( \frac{x - a}{a} \right) + C\]

∫ √ 2 a X − X 2 D X - Mathematics

Question

\int \sqrt{2 a x - x^{2}} dx

Sum

Solution

$Let I = \int \sqrt{2 a x - x^{2}} dx$
$= \int \sqrt{a^{2} + 2 a x - x^{2} - a^{2}} dx$
$= \int \sqrt{a^{2} - (x^{2} - 2 a x + a^{2})} dx$
$= \int \sqrt{a^{2} - {(x - a)}^{2}} dx$
$= (\frac{x - a}{2}) \sqrt{2 a x - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} (\frac{x - a}{a}) + C$

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Indefinite Integral Problems

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Q 15 Q 14 Q 16

Chapter 19: Indefinite Integrals - Exercise 19.28 [Page 155]

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RD Sharma Mathematics [English] Class 12

Chapter 19 Indefinite Integrals
Exercise 19.28 | Q 15 | Page 155

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∫ √ 2 a X − X 2 D X - Mathematics

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