What is the integral from 1 to 5 of (sqrt(24+x^2))/x ?

The integral from 1 to 5 of (sqrt(24+x^2))/x is -sqrt(6)ln((217-88sqrt(6))/(25))+2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 1 to 5 of (sqrt(24+x^2))/x

\int_{1}^{5} \frac{24 + x ^{2}}{x} d x

- 6 ln (\frac{217 - 88 6}{25}) + 2

Decimal Notation

8.98308 \dots

Solution steps

\int_{1}^{5} \frac{24 + x ^{2}}{x} d x

Apply u-substitution

: \int_{5}^{7} \frac{u ^{2}}{u ^{2} - 24} d u

= \int_{5}^{7} \frac{u ^{2}}{u ^{2} - 24} d u

Factor

u^{2} - 24 : - (- u^{2} + 24)

= \int_{5}^{7} \frac{u ^{2}}{- ( - u ^{2} + 24 )} d u

Take the constant out:

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

= - \int_{5}^{7} \frac{u ^{2}}{- u ^{2} + 24} d u

Apply Integral Substitution

= - \int_{\frac{5}{2 6}}^{\frac{7}{2 6}} \frac{2 6 v ^{2}}{- v ^{2} + 1} d v

Take the constant out:

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

= - 26 \cdot \int_{\frac{5}{2 6}}^{\frac{7}{2 6}} \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 26 \cdot \int_{\frac{5}{2 6}}^{\frac{7}{2 6}} \frac{1}{- v ^{2} + 1} - 1 d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 26 (\int_{\frac{5}{2 6}}^{\frac{7}{2 6}} \frac{1}{- v ^{2} + 1} d v - \int_{\frac{5}{2 6}}^{\frac{7}{2 6}} 1 d v)

\int_{\frac{5}{2 6}}^{\frac{7}{2 6}} \frac{1}{- v ^{2} + 1} d v = \frac{ln ( \frac{217 - 88 6}{25} )}{2}

\int_{\frac{5}{2 6}}^{\frac{7}{2 6}} 1 d v = \frac{1}{6}

= - 26 \frac{ln ( \frac{217 - 88 6}{25} )}{2} - \frac{1}{6}

Simplify

- 26 \frac{ln ( \frac{217 - 88 6}{25} )}{2} - \frac{1}{6} : - 6 ln (\frac{217 - 88 6}{25}) + 2

= - 6 ln (\frac{217 - 88 6}{25}) + 2

\frac{d}{d x} (\frac{ln ( x )}{x ^{8}})

\frac{\partial}{\partial x} (\frac{47 x y}{x - y})

\frac{d}{d x} (sec (x) csc (x))

x, 1 - x, - \frac{x}{2}

What is the integral from 1 to 5 of (sqrt(24+x^2))/x ?
The integral from 1 to 5 of (sqrt(24+x^2))/x is -sqrt(6)ln((217-88sqrt(6))/(25))+2