Mathematics

# Evaluate $\displaystyle\int\limits_4^9 {\dfrac{{dx}}{{\sqrt {\left( {9 - x} \right)\left( {x - 4} \right)} }}}$

##### SOLUTION

Consider the given integral.

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{\left( 9-x \right)\left( x-4 \right)}}}$

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{9x-36-{{x}^{2}}+4x}}}$

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{13x-36-{{x}^{2}}}}}$

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{13x+\dfrac{169}{4}-\dfrac{169}{4}-36-{{x}^{2}}}}}$

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{\dfrac{25}{4}-\left( -13x+\dfrac{169}{4}+{{x}^{2}} \right)}}}$

$I=\int_{4}^{9}{\dfrac{dx}{\sqrt{{{\left( \dfrac{5}{2} \right)}^{2}}-{{\left( \dfrac{13}{2}-x \right)}^{2}}}}}$

Let

$t=\dfrac{13}{2}-x$

$dt=-dx$

Therefore,

$I=-\int_{\dfrac{5}{2}}^{-\dfrac{5}{2}}{\dfrac{dt}{\sqrt{{{\left( \dfrac{5}{2} \right)}^{2}}-{{t}^{2}}}}}$

We know that

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

Therefore,

$I=-\left[ {{\sin }^{-1}}\left( \dfrac{t}{\dfrac{5}{2}} \right) \right]_{\dfrac{5}{2}}^{-\dfrac{5}{2}}$

$I=-\left[ {{\sin }^{-1}}\left( \dfrac{2\left( -\dfrac{5}{2} \right)}{5} \right)-{{\sin }^{-1}}\left( \dfrac{2\left( \dfrac{5}{2} \right)}{5} \right) \right]$

$I={{\sin }^{-1}}\left( 1 \right)-{{\sin }^{-1}}\left( 1 \right)$

$I=0$

Hence, the value is $0$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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