What is the integral from 4 to 5 of x/(4-x^2) ?

The integral from 4 to 5 of x/(4-x^2) is 1/2 ln(4/7)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 4 to 5 of x/(4-x^2)

\int_{4}^{5} \frac{x}{4 - x ^{2}} d x

\frac{1}{2} ln (\frac{4}{7})

Decimal

- 0.27980 \dots

Solution steps

\int_{4}^{5} \frac{x}{4 - x ^{2}} d x

Apply u-substitution

= \int_{- 12}^{- 21} - \frac{1}{2 u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 21}^{- 12} - \frac{1}{2 u} d u

Take the constant out:

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

= - (- \frac{1}{2} \cdot \int_{- 21}^{- 12} \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- \frac{1}{2} [ln ∣ u ∣]_{- 21}^{- 12})

Simplify

= \frac{1}{2} [ln ∣ u ∣]_{- 21}^{- 12}

Compute the boundaries:

ln (12) - ln (21)

= \frac{1}{2} (ln (12) - ln (21))

Simplify

= \frac{1}{2} ln (\frac{4}{7})

What is the integral from 4 to 5 of x/(4-x^2) ?
The integral from 4 to 5 of x/(4-x^2) is 1/2 ln(4/7)