Question

Integration

Indefinite Integration

∫ 1/ 1 + x + x² dx

Solve Using TYPE -2

Answer 1

Answer :

Now,

$\int \frac{dx}{1 + x + {x}^{2} } \\ \\ = \int \: \frac{dx}{ {(x + \frac{1}{2} )}^{2} + {( \frac{ \sqrt{3} }{2} )}^{2} } \\ \\ = \frac{1}{ \frac{ \sqrt{3} }{2} } \: {tan}^{ - 1} (\frac{x + \frac{1}{2} }{ \frac{ \sqrt{3} }{2} } ) + c, \\ \\ where \: \: c \: \: is \: \: integral \: \: constant \\ \\ = \frac{2}{ \sqrt{3} } \: {tan}^{- 1} (\frac{2x + 1}{ \sqrt{13} } ) + c \\ \\ RULE : \\ \\ \int \frac{dx}{ {a}^{2} + {x}^{2} } \\ \\ = \frac{1}{a} \: {tan}^{ - 1} ( \frac{x}{a} ) + c, \\ \\ where \: \: c \: \: is \: \: integral \: \: constant$

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