Consider the definite integral ∫[1 0]x^(2)\sqrt{ 4x+8}dx. Find u, f(x), g(u), a, b, and the value of the definite integral.

Question

u=?

f(x)=?

g(u)=?

a=?

b=?

value of the definite integral=?

Answer 1

While we could let u = 4x+8, let's let u = x+2. Then, x = u - 2, and dx = du

Then, if f(x) = x^2 √(4x+8), g(u) = 2(u-2)^2 √u = 2u^5/2 - 8u^3/2 + 8u^1/2

As the original integral has x from 0 to 1, and u = x+2, a = 0+2 = 2 b = 1+2 = 3∫

∫₂³ 2u^5/2 - 8u^3/2 + 8u^1/2 =

4/7u^7/2 - 16/5u^5/2 + 16/3u^3/2 |₂³ = 4√3(15*9-4*21*3+4*35)/35 - 8√2(15*4-4*21*2+4*35)/105 =

92/35 √3 - 256/105 √2 = 1.1048