Question

Evaluate the integral
$\sqrt{5 - 4x - 2x {}^{2} } dx$
​

Answer 1

GIVEN:-

$I = \int \: \frac{dx}{ \sqrt{5 - 4x - 2x {}^{2} } }$

SOLUTION:-

$\implies \: \: I = \int \: \frac{dx}{ \sqrt{ - 2(x {}^{2} + 2x - \frac{5}{2} })} \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - (x {}^{2} + 2x - \frac{5}{2} + 1 - 1)} } \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx }{ \sqrt{ - ((x {}^{2} + 2x + 1) - \frac{5}{2} - 1) } } \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - ((x + 1) {}^{2} - ( \frac{5}{2} + 1)) } } \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - ((x + 1) {}^{2} - \frac{7}{2}) } } \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{( \frac{ \sqrt{7} }{ \sqrt{2} } ) {}^{2} - (x + 1) {}^{2} } } \\ \\ \\ \implies \: \frac{1}{ \sqrt{2} } \sin {}^{ - 1} ( \frac{ 1+x}{ \sqrt{ \frac{7}{2} } } ) + c \\ \\ \\ \rightarrow \: we \: know \: ✪ \boxed{ \int \: \frac{dx}{ \sqrt{a {}^{2} - x {}^{2} } } = \sin {}^{ - 1} ( \frac{x}{a} ) } \\ \\ \\ \implies \: I = \frac{1}{ \sqrt{2} } \sin {}^{ - 1} ( \frac{ \sqrt{2} (x - 1)}{ \sqrt{7} }).$

Hence , the Answer is ,

$\boxed{I = \frac{1}{ \sqrt{2} } \sin {}^{ - 1} ( \frac{ \sqrt{2}(x - 1)}{ \sqrt{7} } ) }$

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HOPE IT HELPS :)

Answer 2

$\bf \underline \blue{Question : - }$

$1)\sf \: evaluate \: \: the \: integral \\ \\ \bf \large \: I = \int \: \frac{dx}{ \sqrt{5 - 4x - {2x}^{2} } }$

$\bf \underline \blue{Answer : - }$

$\bf \large \: I = \int \: \frac{dx}{ \sqrt{5 - 4x - {2x}^{2} } } \\ \\ \implies \large \bf \int \: \frac{dx}{ \sqrt{ - 2( {x}^{2} + 2x - \frac{5}{2}) } } \\ \\ \implies \large \bf \: \frac{1}{ \sqrt{2} } \int \frac{dx}{ \sqrt{ - x( {x}^{2} + 2x - \frac{5}{2} + 1 - 1) } }$

$\bf \large \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - \left[ ({x}^{2} + 2x + 1) \: - \frac{5}{2} - 1 \right]}} \\ \\ \bf \large \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - \left[ {(x + 1)}^{2} - \left( \dfrac{5}{2} + 1 \right) \right]} }$

$\bf \large \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ - \left [ {(x + 1)}^{2} - \frac{\sqrt{{7}}}{2} \right]} } \\ \\ \bf \large \implies \: \frac{1}{ \sqrt{2} } \int \: \frac{dx}{ \sqrt{ \left[ \dfrac{7}{2} ^{2} \right] - {(x + 1)}^{2} } }$

$\bf \large \implies \: \frac{1}{ \sqrt{2} } \: \: {sin}^{ - 1} \left[ \dfrac{x + 1}{ \sqrt{ \frac{7}{2} } } \right] + C \\ \\ \bf \large \: \left[ \because \int \: \frac{dx}{ {a}^{2} - {x}^{2} } = {sin}^{ - 1} \: \frac{x}{a} \right]$

$\bf \large \implies \red{ \frac{1}{ \sqrt{2} } \: {sin}^{ - 1} \left[ \dfrac{ \sqrt{2}(x - 1) }{\sqrt{7}} \right] + C}$