Question

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

Answer 1

Answer:

Convergent; $\frac{1}{32}$.

Step-by-step explanation:

We have been given an integral as $\int _3^{\infty }\:\:\frac{1}{\left(x+1\right)^3}\:dx$. We are asked to determine whether our given integral converges or diverges.

Let us integrate our given integral by u substitution as:

$\int _3^{\infty }\:\:\frac{1}{\left(u)^3}\:dx$

$\int _3^{\infty }\:\:u^{-3}\:dx$

$\int _3^{\infty }\:\:u^{-3}\:dx=\frac{u^{-3+1}}{-3+1}$

$\int _3^{\infty }\:\:u^{-3}\:dx=\frac{u^{-2}}{-3+1}$

$\frac{(x+1)^{-2}}{-2}=-\frac{1}{2(x+1)^2}$

Now, we will compute the boundaries.

$-\frac{1}{2(\infty+1)^2}=-\frac{1}{\infty ^2}=0$

$-\frac{1}{2(3+1)^2}=-\frac{1}{2(4^2}=-\frac{1}{2*16}=-\frac{1}{32}$

Our definite integral would be $0-(-\frac{1}{32})=\frac{1}{32}$

Therefore, our given integral is convergent and its value is $\frac{1}{32}$.

Answer 2

Answer:

$I=\dfrac{1}{32}$

Step-by-step explanation:

Given,

Improper Integral I is given as  

$I=\int^{\infty}_{3}\frac{1}{(x+1)^3}dx$

integration of

$\dfrac{1}{(x+1)^2}$

now 

$\int \dfrac{1}{(x+1)^3}dx = \dfrac{(x+1)^{-3+1}}{-3+1}$

$\int \dfrac{1}{(x+1)^3}dx =\dfrac{1}{-2(x+1)^2}$

$I=\left [\dfrac{1}{-2(x+1)^2}\right]^{\infty}_3$

substituting value  

$I=-\dfrac{1}{2}\left [ \dfrac{1}{\infty^2}-\dfrac{1}{4^2}\right ]$

$I=-\dfrac{1}{2}\left [-\dfrac{1}{16}\right ]$

$I=\dfrac{1}{32}$

so the value of integral converges at $\frac{1}{32}$