Question

Q1.integration of √ ( a² - x² ) dx is?​​

❥Anushka࿐​​
​

Answer 1

Step-by-step explanation:

We have,

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

$\tt\purple{Put\:\:x=a\sin(\theta)}$

$\tt\purple{\implies\:dx=a\cos(\theta)\:d\theta}$

So,

$\int \sqrt{ {a}^{2} - {a}^{2} \sin^{2} ( \theta) } .a \cos( \theta) \: d \theta$

$= a\int \sqrt{ {a}^{2} (1- \sin^{2} ( \theta)) }. \cos( \theta) \: d \theta$

$= a\int \: a \sqrt{1- \sin^{2} ( \theta) }. \cos( \theta) \: d \theta$

$= a^{2} \int \: \cos( \theta) . \cos( \theta) \: d \theta$

$= a^{2} \int \: \cos^{2} ( \theta) \: d \theta$

$= a^{2} \int \: \frac{ 1 + \cos ( 2\theta)}{2} \: d \theta$

$= \frac{a^{2}}{2} \int \: \{ 1 + \cos ( 2\theta) \} \: d \theta$

$= \frac{a^{2}}{2} \int \: 1. d\theta+ \frac{ {a}^{2} }{2} \int \cos ( 2\theta) \: d \theta$

$= \frac{a^{2}}{2} \int \: d\theta+ \frac{ {a}^{2} }{2} \int \cos ( 2\theta) \: d \theta$

$= \frac{a^{2}}{2} .\theta+ \frac{ {a}^{2} }{2} . \frac{\sin ( 2\theta)}{2} + C$

$= \frac{a^{2}}{2} .\theta+ \frac{ {a}^{2} }{2} . \frac{2\sin ( \theta) \cos( \theta) }{2} + C$

$= \frac{a^{2}}{2} .\theta+ \frac{ {a}^{2} }{2} . \sin ( \theta) \cos( \theta) +C$

Now, simce, $x=a\sin(\theta)$, so,

$\theta=\sin^{-1}\bigg(\frac{x}{a}\bigg)$

So, the required integral becomes

$= \frac{a^{2}}{2} . \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2} . \sin \bigg\{ \sin ^{ - 1} \bigg(\frac{x}{a} \bigg) \bigg\} \cos \bigg \{ \sin ^{ - 1} \bigg( \frac{x}{a} \bigg) \bigg \} +C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2}. \bigg(\frac{x}{a} \bigg) . \sqrt{1 - \sin^{2} \bigg \{ \sin ^{ - 1} \bigg( \frac{x}{a} \bigg) \bigg \} }+C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2}. \bigg(\frac{x}{a} \bigg) . \sqrt{1 - \bigg( \frac{x}{a} \bigg)^{2} }+C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2}. \bigg(\frac{x}{a} \bigg) . \sqrt{1 - \frac{x^{2} }{a^{2} } }+C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2}. \bigg(\frac{x}{a} \bigg) . \sqrt{ \frac{a^{2} - x^{2} }{a^{2} } }+C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ {a}^{2} }{2}. \bigg(\frac{x}{a^{2} } \bigg) . \sqrt{ a^{2} - x^{2} }+C$

$= \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ \frac{ x }{2}\sqrt{ a^{2} - x^{2} }+C$

$= \sf \: \red{ \frac{ x }{2}\sqrt{ a^{2} - x^{2} } + \frac{a^{2}}{2} \sin^{ - 1} \bigg( \frac{x}{a} \bigg)+ C }$