Question

Content Quality Answer Required

Solve Using - Indefinite Integrals TYPE - 8

Question = ∫ 2 x + 3 / √4x + 3 dx

Answer 1

Answer :

Let, 4x + 3 = z²

∴ 4 dx = 2z dz

or, dx = (z/2) dz

Also, 2x + 3

= (z² - 3)/2 + 3

= (z² - 3 + 6)/2

= (z² + 3)/2

Now,

$\int \frac{(2x + 3)dx}{ \sqrt{4x + 3} } \\ \\ = \int \frac{( \frac{ {z}^{2} + 3}{2}) \: (\frac{z}{2} \: dz)}{z} \\ \\ = \frac{1}{4} \int( {z}^{2} + 3) \: dz \\ \\ = \frac{1}{4} \int {z}^{2} \: dz + \frac{3}{4} \int \: dz \\ \\ = \frac{1}{4} \: \frac{{z}^{3} }{3} + \frac{3z}{4} \: + c, \\ \\ where \: \: c \: \: is \: \: integral \: \: constant \\ \\ = \frac{ {z}^{3} }{12} + \frac{3z}{4} + c \\ \\ = \frac{ {(4x + 3)}^{ \frac{3}{2} } }{12} + \frac{3 \sqrt{4x + 3} }{4} + c\\ \\ = \frac{ \sqrt{(4x + 3)} }{4} ( \frac{4x + 3}{3} + 3 ) + c\\ \\ = \frac{ \sqrt{(4x + 3)} }{4} ( \frac{4x + 3 + 9}{3} ) + c \\ \\ = \frac{ \sqrt{(4x + 3)} }{4} ( \frac{4x + 12}{3} ) + c \\ \\ = \frac{(x + 3)}{3} \sqrt{(4x + 3)} + c$

∵ z = √(4x + 3)

#MarkAsBrainliest

Answer 2

Dear Student,

please find the solution in attachment.