Find out the answer to "Decide whether the integral is improper. ∫12dxx3"

Cheyanne Leigh

Answered question

2021-11-08

Decide whether the integral is improper.

\int_{1}^{2} \frac{d x}{x^{3}}

Answer & Explanation

Clelioo

Skilled2021-11-09Added 88 answers

Step 1
Given- $\int_{1}^{2} \frac{d x}{x^{3}}$
To decide- Whether the given integeral is improper.
Concept Used- Using the power rule as $\int_{a}^{b} x^{n} d x = {[\frac{x^{n + 1}}{n + 1}]}_{a}^{b}$ , we can solve the above integration.
Step 2
Explanation- Rewrite the given integral as,
$I = \int_{1}^{2} \frac{d x}{x^{3}}$
Solving further, we get,
$= \int_{1}^{2} x^{- 3} d x$
$= {[\frac{x^{- 3 + 1}}{- 3 + 1}]}_{1}^{2}$
$= \frac{1}{- 2} {[\frac{1}{x^{2}}]}_{1}^{2}$
$= \frac{- 1}{2} [\frac{1}{4} - 1]$
$= \frac{- 1}{2} [\frac{- 3}{4}]$
$= \frac{3}{8} < 1$
As the value of the above integral is $\frac{3}{8}$ , so the above integral is proper integral.
Answer- Hence, the integral $\int_{1}^{2} \frac{d x}{x^{3}}$ is proper integral and its value is $\frac{3}{8}$ .

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