Question

Integral 3/x^5 dx over (0,1]

Answer 1

Answer: We have to solve the following definite integral over a specified interval:

$\int_0^1\frac{3}{x^5}dx$

The solution steps are as follows:

$\begin{gathered} \int_0^1\frac{3}{x^5}dx=3\int_0^1\frac{1}{x^5}dx\Rightarrow(1) \\ \\ \end{gathered}$

Applying the reverse power rule on the (1) gives the following answer:

$\begin{gathered} 3\int_0^1\frac{1}{x^5}dx=3[\int_0^1\frac{1}{x^5}dx=-\frac{1}{4x^4}] \\ \\ \\ \int_0^1\frac{3}{x^5}dx=[-\frac{1}{4x^4}]\rightarrow\text{ Evaluated at \lparen0,1\rparen} \\ \\ ----------------------- \\ -\frac{1}{4x^4}\rightarrow-\frac{1}{4(1)^4}-(-\frac{1}{4(0)^4}) \\ \\ \text{ Do note!} \\ (-\frac{1}{4(0)^4})\rightarrow\text{ Become infiniate} \\ \\ \text{ Therefore.} \\ \\ \\ -\frac{1}{4(1)^4}-(-\frac{1}{4(0)^4})=\text{ Diverges.} \\ \\ \text{ So the answer is:} \\ \\ \int_0^1\frac{3}{x^5}dx=\text{ Diverges} \end{gathered}$

The integral therefore diverges.

The plot of the parent function clearly shows the behavior between (0,1)