Mathematics

Evaluate the following integrals:
$$\int { \sqrt { 2ax-{ x }^{ 2 } }  } dx\quad $$


SOLUTION
Let $$I=\displaystyle\int \sqrt{2 a x-x^2}\ d x$$

$$\implies I=\displaystyle\int \sqrt{-(-2(a)(x)+x^2)}\ d x$$

$$\implies I=\displaystyle\int \sqrt{a^2-(a^2-2(a)(x)+x^2}\ d x$$

$$\implies I=\displaystyle\int \sqrt{a^2-(a-x)^2}\ d x$$

Put $$t=a-x\implies d t=-d x$$

$$\implies I=-\displaystyle\int \sqrt{a^2-t^2}\ d t$$

As we know that

$$\displaystyle\int \sqrt{a^2-x^2}\ d x=\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{x}{a}\right)+C$$

$$I=-\dfrac{t}{2}\sqrt{a^2-t^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{t}{a}\right)+C$$

$$I=-\dfrac{a-x}{2}\sqrt{a^2-(a-x)^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{a-x}{a}\right)+C$$

$$I=\dfrac{1}{2}(x-a)\sqrt{2 a x-x^2}-\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{a-x}{a}\right)+C$$
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Subjective Medium Published on 17th 09, 2020
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