Best solutions to - "Evaluate the definite integral. ∫12e1x4x5dx"

Priscilla Johnston

Answered question

2021-12-21

Evaluate the definite integral.

\int_{1}^{2} \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x

Answer & Explanation

Anzante2m

Beginner2021-12-22Added 34 answers

Step 1
To evaluate the definite integral.
Step2
Given information:
$\int_{1}^{2} (\frac{e^{\frac{1}{x^{4}}}}{x^{5}}) d x$
Step 3
Calculation:
Integrate the function with respect to x.
$\int_{1}^{2} (\frac{e^{\frac{1}{x^{4}}}}{x^{5}}) d x$
Let $\frac{1}{x^{4}} = u$
so, differentiate with respect to x.
$\frac{d}{d x} (\frac{1}{x^{4}}) = \frac{d}{d x} (u)$
$\frac{- 4}{x^{5}} = \frac{d u}{d x}$
$(\frac{1}{x^{5}}) d x = \frac{1}{- 4} d u$
[put in the given integral]
Step 4
So, the integral become,
$\int_{x = 1}^{2} (\frac{e^{\frac{1}{x^{4}}}}{x^{5}}) d x = \int_{x = 1}^{2} (\frac{e^{u}}{- 4}) d u$
$[p u t (\frac{1}{x^{5}}) d x = (\frac{1}{x^{- 4}}) d u, and u = \frac{1}{x^{4}}]$
$= - \frac{1}{4} {[e^{u}]}_{x = 1}^{2}$
$[∵ \int e^{x} d x = e^{x}]$
$= - \frac{1}{4} {[e^{\frac{1}{x^{4}}}]}_{x = 1}^{2}$
$[s u b s t i t u t e u = \frac{1}{x^{4}}]$
$= - \frac{1}{4} [e^{\frac{1}{2^{(4)}}} - e^{\frac{1}{1^{(4)}}}]$

limacarp4

Beginner2021-12-23Added 39 answers

Step 1
Let

u = \frac{1}{x^{4}}

so that

d u = - \frac{4}{x^{5}} d x

By substitution,

\int \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = \int e^{u} (- \frac{1}{4} d u)

(1)

\int \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = - \frac{1}{4} \int e^{u} d u

Recall:

\int e^{u} d u = e^{u} + C

so:

\int \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = - \frac{1}{4} e^{u} + C

\int \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = - \frac{1}{4} e^{\frac{1}{x^{4}}} + C

Hence,

\int_{1}^{2} \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = {[- \frac{1}{4} e^{\frac{1}{x^{4}}}]}_{1}^{2}

= - \frac{1}{4} e^{\frac{1}{2^{4}}} - (- \frac{1}{4} e^{\frac{1}{1^{4}}})

= - \frac{1}{4} e^{\frac{1}{16}} + \frac{1}{4} e

= \frac{1}{4} (- e^{\frac{1}{16}} + e)

\approx 0.413

nick1337

Expert2021-12-28Added 777 answers

Step 1
We have to calculate
$\int_{1}^{2} \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x$
Let us first calculate indefinite integral.
Given integral
Step 2
$\int \frac{e^{\frac{1}{4^{4}}}}{x^{5}} d x = - \frac{1}{4} \int e^{u} d u$
Substitute $u = \frac{1}{x^{4}}$ this gives us
$d x = - \frac{x^{5}}{4} d u$
Step 3
$- \frac{1}{4} \int e^{u} d u = - \frac{e^{u}}{4} = - \frac{e^{\frac{1}{x^{4}}}}{4}$
Substitute back $u = \frac{1}{x^{4}}$
Step 4
$\int \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = - \frac{e^{\frac{1}{4^{5}}}}{4} + C$
$\int_{1}^{2} \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x = - [\frac{e^{\frac{1}{4^{5}}}}{4}]_{1}^{2}$
$= \frac{e}{4} - \frac{\sqrt[16]{e}}{4}$

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