Question

Integration by parts

Answer 1

Answer :

Now,

$\int \sqrt{4 - 3x - 2 {x}^{2} } \: dx \\ \\ = \sqrt{2}\int \sqrt{2 - \frac{3}{2}x - {x}^{2} } \: dx \\ \\ = \sqrt{2} \int \sqrt{2 + \frac{9}{16} - {(x + \frac{3}{4} )}^{2} } \: dx \\ \\ = \sqrt{2} \int \sqrt{ { (\frac{ \sqrt{41}}{4} ) }^{2} - {(x + \frac{3}{4} )}^{2} } \: dx \\ \\ = \sqrt{2} \frac{(x + \frac{3}{4} ) \sqrt{ {( \frac{ \sqrt{41} }{4} )}^{2} - {(x + \frac{3}{4} )}^{2} } }{2} \\ + \sqrt{2} \frac{ ({ \frac{ \sqrt{41} }{4} )}^{2} }{2} {sin}^{ - 1} ( \frac{x + \frac{3}{4} }{ \frac{ \sqrt{41} }{4} } ) + c \\ \\ where \: c \: is \: integral \: constant \\ \\ = \frac{ \sqrt{2}(4x + 3) \sqrt{4 - 3x - 2 {x}^{2} } }{32} \\ + \frac{41 \sqrt{2} }{32} {sin}^{ - 1} \frac{4x + 3}{ \sqrt{41} } + c$

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