Question

Content Quality Answer Required

Solve using - Indefinite Integrals TYPE - 8

Question = ∫ 1 / ( 2x + 3 ) √x + 2 dx

Answer 1

Answer :

Let, x + 2 = z²

∴ dx = 2z dz

Also, 2x + 3

= 2z² - 4 + 3

= 2z² - 1

Now,

$\int \frac{dx}{(2x + 3) \sqrt{x + 2} } \\ \\ = \int \frac{2z \: dz}{(2 {z}^{2} - 1)z } \\ \\ = \int \frac{dz}{ {(z)}^{2} - {( \frac{1}{ \sqrt{2} }) }^{2} } \\ \\ = \frac{1}{2 (\frac{1}{ \sqrt{2} }) } \: log | (\frac{z - \frac{1}{ \sqrt{2} } }{z + \frac{1}{ \sqrt{2} } }) | + c, \\ \\ where \: \: c \: \: is \: \: integral \: \: constant \\ \\ = \frac{1}{ \sqrt{2} } \: log | \frac{ \sqrt{x + 2} - \frac{1}{ \sqrt{2} } }{ \sqrt{x + 2} + \frac{1}{ \sqrt{2} } } | + c \\ \\ = \frac{1}{ \sqrt{2} } \: log | \frac{ \sqrt{2(x + 2)} - 1}{ \sqrt{2(x + 2)} +1 } | + c$

RULE :

$\int \frac{dx}{ {x}^{2} - {a}^{2} } \\ \\ = \frac{1}{2a} \: log | \frac{x - a}{x + a} | \: + c, \\ \\ where \: \:c \: \: is \: \: integral \: \: constant$

#MarkAsBrainliest