Either evaluate the given improper integral or show that it diverges. `int_1^(+oo) x/(x+1)^2 dx`

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lemjay | High School Teacher | (Level 2) Senior Educator

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`int_1^(+oo) x/(x+1)^2 dx`

Decompose the integrand into its equivalent partial fractions.

`x/(x+1)^2 = A/(x+1)+B/(x+1)^2`

Multiply both sides by the LCD `(x+1)^2` .

`x = A(x+1) + B`

To solve for B, set x=-1.

`-1=A(-1+1)+B `          `-1=A(0)+B`          `B=-1`

To solve for A set x=0 and substitute B=-1.

`0=A(0+1)+(-1)`              `0=A-1`                  `A=1`

So we have,

`int_1^(x) x/(x+1)^2dx =int_1^(+oo) (1/(x+1)-1/(x+1)^2) dx = int_1^(+oo) 1/(x+1)dx - int_1^(+oo)1/(x+1)^2dx`

Since the upper limit of the integral is infinity, replace the infinity with t variable to be able to evaluate the integral. 

`= int_1^t 1/(x+1)dx - int_1^t 1/(x+1)^2dx`

Then, let's try to evaluate the two integrals separately.

For the first integral, apply the logarithm formula which is `int1/udu=lnu+C` .

`int_1^t 1/(x+1)dx = ln(x+1)`  `|_1^t = ln (t+1) - ln 2`

Note that t represents +infinity, so take the limit of t as it approaches +infinity, to evaluate further.

To do so, apply the rule for logarithm which is `lim_(x->oo) lnx= oo`

`lim_(t->+oo)`  `[ln(t+1) - ln2] = oo-ln2 = oo`

The limit is infinity because if we subtract a large number with a small one, the resulting value is a large number.

Next, let's evaluate the other integral using u-substitution.

`int_1^t 1/(x+1)^2 dx`

`u = x+1`        and       `du=dx`

`int 1/u^2 du = int u^(-2)du= u^(-1) = 1/u`

Then, substitute back u=x+1.

`int_1^t 1/(x+1)^2 dx = 1/(x+1)`  `| _1^t = 1/(t+1) - 1/(1+1) = 1/(t+1)-1/2`

Then, take the limit of t as it approaches +infinity. Note that` lim_(x->oo) 1/x = 0` .

 `lim_(t->+oo)`  `[1/(t+1) - 1/2 ]= 0- 1/2 = -1/2`                                                       

Next substitute the limits of the two integral. So we have,

`int_1^(+oo) x/(x+1)^2 dx = int_1^(+oo) 1/(x+1) dx - int_1^(+oo)1/(x+1)^dx = oo + 1/2 = oo`

Note that if a small number is added to a large number, the sum has a large value.

Hence, `int_1^oo x/(x+1)^2dx = oo` .

Since we have infinity, the integral diverges.


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