What is the integral from-1 to 0 of 3(e^{1/x})/(x^3) ?

The integral from-1 to 0 of 3(e^{1/x})/(x^3) is -6/e

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from-1 to 0 of 3(e^{1/x})/(x^3)

\int_{- 1}^{0} 3 \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

- \frac{6}{e}

Decimal Notation

- 2.20727 \dots

Solution steps

\int_{- 1}^{0} 3 \cdot \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{- 1}^{0} \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Apply Integration By Parts

: [\frac{e ^{\frac{1}{x}}}{x ^{2}} - \int - \frac{e ^{\frac{1}{x}} ( 3 x + 1 )}{x ^{4}} d x]_{- 1}^{0}

= 3 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - \int - \frac{e ^{\frac{1}{x}} ( 3 x + 1 )}{x ^{4}} d x]_{- 1}^{0}

\int - \frac{e ^{\frac{1}{x}} ( 3 x + 1 )}{x ^{4}} d x = \frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x}

= 3 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - (\frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x})]_{- 1}^{0}

Simplify

3 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - (\frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x})]_{- 1}^{0} : 3 [\frac{- e ^{\frac{1}{x}} + e ^{\frac{1}{x}} x}{x}]_{- 1}^{0}

= 3 [\frac{- e ^{\frac{1}{x}} + e ^{\frac{1}{x}} x}{x}]_{- 1}^{0}

Compute the boundaries:

- \frac{2}{e}

= 3 (- \frac{2}{e})

Simplify

= - \frac{6}{e}

\frac{d}{d x} (\frac{1}{π} sin (π x))

x \to 5 lim (10 x + 35)

\frac{d}{d x} (ln (arcsin (x^{2})))

\int (x^{3} - 6 x) d x

\frac{d}{d x} (x^{3} - 6 x^{2} - 15 x + 40)

What is the integral from-1 to 0 of 3(e^{1/x})/(x^3) ?
The integral from-1 to 0 of 3(e^{1/x})/(x^3) is -6/e