What is the integral of (sqrt(x))/(x-81) ?

The integral of (sqrt(x))/(x-81) is -18(1/2 ln|(sqrt(x))/9+1|-1/2 ln|(sqrt(x))/9-1|-1/9 sqrt(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sqrt(x))/(x-81)

\int \frac{x}{x - 81} d x

- 18 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1 - \frac{1}{9} x) + C

Solution steps

\int \frac{x}{x - 81} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{2} - 81} d u

= \int \frac{2 u ^{2}}{u ^{2} - 81} d u

Take the constant out:

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

= 2 \cdot \int \frac{u ^{2}}{u ^{2} - 81} d u

Factor

u^{2} - 81 : - (- u^{2} + 81)

= 2 \cdot \int \frac{u ^{2}}{- ( - u ^{2} + 81 )} d u

Take the constant out:

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

= 2 (- \int \frac{u ^{2}}{- u ^{2} + 81} d u)

Apply Integral Substitution

= 2 (- \int \frac{9 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

= 2 (- 9 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v)

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

= 2 (- 9 \cdot \int \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

= 2 (- 9 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v))

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= 2 (- 9 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Substitute back

= 2 - 9 \frac{ln \frac{x}{9} + 1}{2} - \frac{ln \frac{x}{9} - 1}{2} - \frac{x}{9}

Simplify

2 - 9 \frac{ln \frac{x}{9} + 1}{2} - \frac{ln \frac{x}{9} - 1}{2} - \frac{x}{9} : - 18 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1 - \frac{1}{9} x)

= - 18 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1 - \frac{1}{9} x)

Add a constant to the solution

= - 18 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1 - \frac{1}{9} x) + C

\int \frac{( 3 + 2 y ) ^{2}}{y ^{5}} d y

\int 2 x y^{3} + y^{6} d x

\int \frac{x ^{2}}{x ^{3} + 10} d x

\int \frac{2}{5} - \frac{1}{25} x d x

\int sin (x) - 3 cos (x) d x

What is the integral of (sqrt(x))/(x-81) ?
The integral of (sqrt(x))/(x-81) is -18(1/2 ln|(sqrt(x))/9+1|-1/2 ln|(sqrt(x))/9-1|-1/9 sqrt(x))+C