What is the integral of (18)/(xsqrt(1+2x)) ?

The integral of (18)/(xsqrt(1+2x)) is -36(1/2 ln|sqrt(1+2x)+1|-1/2 ln|sqrt(1+2x)-1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (18)/(xsqrt(1+2x))

\int \frac{18}{x 1 + 2 x} d x

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

Solution steps

\int \frac{18}{x 1 + 2 x} d x

Take the constant out:

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

= 18 \cdot \int \frac{1}{x 1 + 2 x} d x

Apply u-substitution

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

= 18 \cdot \int \frac{2}{u ^{2} - 1} d u

Take the constant out:

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

= 18 \cdot 2 \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

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

= 18 \cdot 2 \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 18 \cdot 2 (- \int \frac{1}{- u ^{2} + 1} d u)

Use the common integral:

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

= 18 \cdot 2 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = 1 + 2 x

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

Simplify

18 \cdot 2 (- (\frac{ln 1 + 2 x + 1}{2} - \frac{ln 1 + 2 x - 1}{2})) : - 36 (\frac{1}{2} ln 1 + 2 x + 1 - \frac{1}{2} ln 1 + 2 x - 1)

= - 36 (\frac{1}{2} ln 1 + 2 x + 1 - \frac{1}{2} ln 1 + 2 x - 1)

Add a constant to the solution

= - 36 (\frac{1}{2} ln 1 + 2 x + 1 - \frac{1}{2} ln 1 + 2 x - 1) + C

x \to 1 + lim (\frac{x + 1}{x - 1})

y^{^{''}} - 5 y^{^{'}} + 4 y = 0

\int \frac{6 x}{x ^{2} + 1} d x

\frac{d ^{2}}{d x ^{2}} ((6 x^{2} + 8)^{6})

n = 1 \sum \infty \frac{4}{π ^{n}}

What is the integral of (18)/(xsqrt(1+2x)) ?
The integral of (18)/(xsqrt(1+2x)) is -36(1/2 ln|sqrt(1+2x)+1|-1/2 ln|sqrt(1+2x)-1|)+C