Question

Integration of (x+2)underoot 3x+5

Answer 1

Solution :

Let, 3x + 5 = z²

⇒ 3 dx = 2z dz

⇒ dx = (2z/3) dz

$\therefore\mathsf{\int (x+2)\sqrt{3x+5}\:dx}$

= ∫ {(z² - 5)/3 + 2} z (2z/3) dz

= 2/3 ∫ (z² - 5 + 6)/3 z² dz

= 2/9 ∫ (z² + 1) z² dz

= 2/9 ∫ (z⁴ + z²) dz

= 2/9 ∫ z⁴ dz + 2/9 ∫ z² dz

= 2/9 z⁵/5 + 2/9 z³/3 + C ,

where C is integral constant

= 2/45 z⁵ + 2/27 z³ + C

= $\frac{2}{45}(3x+5)^{5/2}+ \frac{2}{27}(3x+5)^{3/2}$ + C

= $\frac{2}{45}\sqrt[5]{3x+5}+ \frac{2}{27}\sqrt[3]{3x+5}$ + C ,

which is the required integral solution