Question

Integration.

Indefinite Integration.

∫1/4x² - 4x - 7

Solve using TYPE - 1

Answer 1

Answer :

Now,

$\int \frac{dx}{4 {x}^{2} - 4x - 7 } \\ \\ = \int \frac{dx}{(4 {x}^{2} - 4x + 1) - 8} \\ \\ = \int \frac{dx}{ {(2x - 1)}^{2} - {(2 \sqrt{2}) }^{2} } \\ \\ = \frac{1}{2} \int \frac{dz}{ {z}^{2} - {(2 \sqrt{2} )}^{2} } \\ \\ - - - - - - - - - \\ Let,\: \: 2x - 1 = z \\ \\ \implies2 \: dx = dz \\ \\ \implies \: dx = \frac{dz}{2} \\ - - - - - - - - - \\ \\ = \frac{1}{2} \frac{1}{2 ( 2 \sqrt{2}) } \: log | (\frac{z - 2 \sqrt{2} }{z + 2 \sqrt{2} }) | + c, \\ \\ where \: \: c \: \: is \: \: integral \: \: constant \\ \\ = \frac{1}{8 \sqrt{2} } \: log |( \frac{2x - 1 - 2 \sqrt{2} }{2x - 1 + 2 \sqrt{2} } )| + c \\ \\ RULE : \\ \\ \int \frac{dx}{ {x}^{2} - {a}^{2} } \\ \\ = \frac{1}{2a} \: log | \frac{x - a}{x + a} | + c, \\ \\ where \: \: c \: \: is \: \: inegral \: \: constant$

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