What is the integral of (3sqrt(x))/(x-1) ?

The integral of (3sqrt(x))/(x-1) is -6(1/2 ln|sqrt(x)+1|-1/2 ln|sqrt(x)-1|-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 (3sqrt(x))/(x-1)

\int \frac{3 x}{x - 1} d x

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

Solution steps

\int \frac{3 x}{x - 1} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{x - 1} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{2} - 1} d u

= 3 \cdot \int \frac{2 u ^{2}}{u ^{2} - 1} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{u ^{2}}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= 3 \cdot 2 \cdot \int \frac{u ^{2}}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 3 \cdot 2 (- \int \frac{u ^{2}}{- u ^{2} + 1} d u)

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

= 3 \cdot 2 (- \int \frac{1}{- u ^{2} + 1} - 1 d u)

Apply the Sum Rule:

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

= 3 \cdot 2 (- (\int \frac{1}{- u ^{2} + 1} d u - \int 1 d u))

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

\int 1 d u = u

= 3 \cdot 2 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2} - u))

Substitute back

u = x

= 3 \cdot 2 (- (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2} - x))

Simplify

3 \cdot 2 (- (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2} - x)) : - 6 (\frac{1}{2} ln x + 1 - \frac{1}{2} ln x - 1 - x)

= - 6 (\frac{1}{2} ln x + 1 - \frac{1}{2} ln x - 1 - x)

Add a constant to the solution

= - 6 (\frac{1}{2} ln x + 1 - \frac{1}{2} ln x - 1 - x) + C

h \to 0 lim (2)

\frac{d P}{d t} = \frac{1}{10} P (500 - P)

\int sec^{2} (5 x) tan^{4} (5 x) d x

x \to 1 lim (\frac{x ^{2} - 4}{x - 1})

f (t) = e^{t + 7}

What is the integral of (3sqrt(x))/(x-1) ?
The integral of (3sqrt(x))/(x-1) is -6(1/2 ln|sqrt(x)+1|-1/2 ln|sqrt(x)-1|-sqrt(x))+C