`int_0^5 x^2/(5+2x)^2 dx` Use integration tables to evaluate the definite integral.

`int_0^5x^2/(5+2x)^2dx`

First let's evaluate the indefinite integral,

Use the following from the integration table:

`intu^2/(a+bu)^2du=1/b^3(bu-a^2/(a+bu)-2aln|a+bu|)+C`

Here we have `a=5,b=2`

`intx^2/(5+2x)^2dx=1/2^3(2x-5^2/(5+2x)-2(5)ln|5+2x|)+C`

`=1/8(2x-25/(5+2x)-10ln|5+2x|)+C`

So, `int_0^5x^2/(5+2x)^2=1/8[2x-25/(5+2x)-10ln|5+2x|]_0^5`

`=1/8{[2(5)-25/(5+2(5))-10ln|5+2(5)|]-[2(0)-25/(5+2(0))-10ln|5+2(0)\]}`

`=1/8{[10-25/15-10ln|15|]-[-25/5-10ln|5|]}`

`=1/8(10-5/3-10ln15+5+10ln5)`

`=1/8(15-5/3-10(ln15-ln5))`

`=1/8(40/3-10ln(15/5))`

`=1/8(40/3)-(10/8)ln3`

`=5/3-5/4ln3`

`~~0.2934`

Comments

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