Question

Determine whether the improper integral converges or diverges, and find the value of each that converges.

Answer 1

Integration is the process of finding integral. The value of the integration $\int_{{{0}}}^{{\infty}}}{\dfrac{{{\left.{d}{x}\right.}}}{{{\left({x}+{9}\right)}^{{{2}}}}}}\)$ converges.

What is integration?

Integration is the process of finding integral. Integral help us to describe the function of area, volume, and other such concepts.

Given to us

$\int_{{{0}}}^{{\infty}}}{\dfrac{{{\left.{d}{x}\right.}}}{{{\left({x}+{9}\right)}^{{{2}}}}}}\)$

To know whether the internal will converge or diverge, we will solve the integral,

$\int_{{{0}}}^{{\infty}}}{\dfrac{{{\left.{d}{x}\right.}}}{{{\left({x}+{9}\right)}^{{{2}}}}}}\)\\\\\\ =[ -\dfrac{1}{x+9}]_0^\infty\\\\= \dfrac{1}{9}\\\\= 0.\overline{1}$

As we can see that the value of the integration $\int_{{{0}}}^{{\infty}}}{\dfrac{{{\left.{d}{x}\right.}}}{{{\left({x}+{9}\right)}^{{{2}}}}}}\)$ converges toward zero, therefore we can say that the given integration converges.

Hence, the value of the integration $\int_{{{0}}}^{{\infty}}}{\dfrac{{{\left.{d}{x}\right.}}}{{{\left({x}+{9}\right)}^{{{2}}}}}}\)$ converges.

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