Use a u-substitution to completely rewrite the definite integral in terms of u. Give exact values only. No decimals. ∫[7 5] sqrt(3x+5)dx=∫[b a]g(u)du

Question

u=?

du=? dx

a=?

b=?

g(u)=?

Answer 1

Given: Integral from x=7 to x=5 of sqrt(3x+5) dx

Use U-sub to simplify

Note: Usually integrals start from a

lower value of x to a larger value of x.

Solution:

u = 3x + 5

du/dx = 3

du = 3 dx

x=7 ==> u = 3(7) + 5 = 26

x=5 ==> u = 3(5) + 5 = 20

I = The Integral

I = Integral from x=7 to x=5 of sqrt(3x+5) dx

I = (1/3) Integral from u=26 to u=20 of sqrt(u) du

Note: The (1/3) corrects for the additional 3 in du.

I = (1/3) [ (2/3)*u^(3/2) ] from u=26 to u=20

I = (1/3)(2/3) [ (26)^(3/2) - (20)^(3/2) ]

I = (2/9) [ (20)^(3/2) - (26)^(3/2) ]

I = -9.5848