# Use integration by parts to find the given integral. Use symbolic notations where needed 8 on top of integral sign and 0 on bottom of integral sign...11x/squareroot(5x+9)  dx

You need to evaluate the definite integral int_0^8 (11x)/(sqrt(5x + 9))dx using integration by parts, such that:

int_0^8 (11x)/(sqrt(5x + 9))dx = (2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx

Considering u = x and dv = 5/(2sqrt(5x+9))dx yields:

u = x => du = dx

dv = 5/(2sqrt(5x+9))dx => v = sqrt(5x+9)

(2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = (2*11)/5*(x*sqrt(5x+9)|_0^8 - int_0^8 sqrt(5x+9) dx)

You need to evaluate the definite integral int_0^8 sqrt(5x+9) dx using substitution, such that:

5x+9 = t => 5dx = dt => dx = (dt)/5

x = 0 => t = 9

x = 8 => t = 49

int_0^8 sqrt(5x+9) dx = int_9^49 sqrt t*(dt)/5

int_9^49 sqrt t*(dt)/5 = (1/5) int_9^49 t^(1/2) dt

(1/5) int_9^49 t^(1/2) dt = (1/5)*(2/3) tsqrt t|_9^49

(1/5) int_9^49 t^(1/2) dt = (2/15)(49*sqrt49 - 9*sqrt9)

(1/5) int_9^49 t^(1/2) dt = (2/15)(49*7 - 9*3)

(1/5) int_9^49 t^(1/2) dt = 632/15

(2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = (2*11)/5*(8sqrt49 - (2/15)*316)

(2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = (2*11)/5*(56 - (2/15)*316)

(2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = (3696 - 632)/15

(2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = (3064)/15 ~~ 204.26

Hence, evaluating the given definite integral, using parts and substitution, yields (2*11)/5 int_0^8 x*5/(2sqrt(5x+9))dx = 204.26.