Best solutions to - "Evaluate the following integrals. Include absolute values only..."

permaneceerc

Answered question

2021-10-12

Evaluate the following integrals. Include absolute values only when needed.

\int_{e}^{e^{2}} \frac{d x}{x \ln^{3} x}

Answer & Explanation

nitruraviX

Skilled2021-10-13Added 101 answers

Given:

\int_{e}^{e^{2}} \frac{d x}{x \ln^{3} x}

To find- The value of the above integral.
Concept Used- Substitution method can be used to find the value of the above integral.
Explanation- Rewrite the given integral,

\int_{e}^{e^{2}} \frac{d x}{x \ln^{3} x}

Let,

I = \int_{e}^{e^{2}} \frac{1}{x} \cdot \frac{d x}{\ln^{3} x}

Now, substitute

\ln x = t

and differentiate both sides with respect to x, we get,

\frac{1}{x} d x = d t

As we have substituted

\ln x = t

, so we have to change the limit which is as follows,
at x=e, so

t = \ln e

t=1
at

x = e^{2}

, so

t = {\ln e}^{2}

t=2
So, the above integrals becomes,

I = \in_{1}^{2} \frac{1}{t^{3}} d t

= \int_{1}^{2} t^{- 3} d t

= {[\frac{t^{- 3 + 1}}{- 3 + 1}]}_{1}^{2}

= {[\frac{t^{- 2}}{- 2}]}_{1}^{2}

= \frac{- 1}{2} \cdot [\frac{1}{2^{2}} - \frac{1}{1^{2}}]

= \frac{- 1}{2} [\frac{1}{4} - 1]

= \frac{- 1}{2} \cdot (\frac{- 3}{4})

= \frac{3}{8}

Answer: Hence, the value of the integral

\int_{e}^{e^{2}} \frac{d x}{x \ln^{3} x}

\frac{3}{8}

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