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Use integration by parts to find the given integral. Use symbolic notations where...

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maheen100 | Student, Undergraduate | (Level 1) Honors

Posted February 21, 2013 at 5:22 PM via web

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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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 21, 2013 at 5:50 PM (Answer #1)

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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 d`v = 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.`

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