We will evaluate the following improper integral

Question

We will evaluate the following improper integral:

∫ x/(x^2 + 4)^2 dx from 3 to ∞

First we will do the indefinite integral ∫ x/(x^2 + 4)^2 dx

(a) Use the substitution u=x^2+4 to formulate as an integral with respect to u, filling in the integrand below (remember to use proper Mobius syntax in your function)

(b)

Now integrate to get a function of u:

(c)

Substitute the value of u in terms of x to get an antiderivative in terms of x, we will call this F(x):

F(x)=

(d) Now to do the improper integral we require

lim a-> ∞ ∫ x/(x^2 + 4)^2 dx from 3 to a

so for this we need

[F(x)]from 3 to a =

(the answer here is a function of a)

e) Finally to evaluate the answer take the limit as a→∞. Enter the exact answer (e.g. as a fraction if necessary, not a decimal).

 

Answer 1

∫ [x / (x²+4)²]dx Let u = x²+4. Then du = 2xdx. So, xdx = (1/2)du

= (1/2)∫ u^-2du = -1 / [2u] + C = -1 / [2(x² + 4)] + C

So, we have ∫_{(from 3 to ∞)} [x / (x²+4)²]dx = (-1/2)lim_a→∞ [1 / (x²+4)]_{(from 3 to a)}

= (-1/2)lim_a→∞ [1 / (a²+4) - 1 / 13] = (-1/2)[0 - 1/13] = 1/26