What is the integral of (x^2)/(5-x^6) ?

The integral of (x^2)/(5-x^6) is 1/(6sqrt(5))(ln|(x^3)/(sqrt(5))+1|-ln|(x^3)/(sqrt(5))-1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2)/(5-x^6)

\int \frac{x ^{2}}{5 - x ^{6}} d x

\frac{1}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1) + C

Solution steps

\int \frac{x ^{2}}{5 - x ^{6}} d x

Apply u-substitution

: \int \frac{1}{3 ( 5 - u ^{2} )} d u

= \int \frac{1}{3 ( 5 - u ^{2} )} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{5 - u ^{2}} d u

Apply Integral Substitution

= \frac{1}{3} \cdot \int \frac{1}{5 ( 1 - v ^{2} )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{5} \cdot \int \frac{1}{1 - v ^{2}} d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{5} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= \frac{1}{3} \cdot \frac{1}{5} \frac{ln \frac{x ^{3}}{5} + 1}{2} - \frac{ln \frac{x ^{3}}{5} - 1}{2}

Simplify

\frac{1}{3} \cdot \frac{1}{5} \frac{ln \frac{x ^{3}}{5} + 1}{2} - \frac{ln \frac{x ^{3}}{5} - 1}{2} : \frac{1}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1)

= \frac{1}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1)

Add a constant to the solution

= \frac{1}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1) + C

\int \frac{1 + tan ( x )}{1 - cos ( x )} d x

\int e^{- x} sin^{2} (2 x) d x

\int (\frac{8}{x} - \frac{5}{x ^{2}} + \frac{6}{x ^{3}}) d x

\int (2 x^{2} + 3 x + 2) sin (3 x) d x

\int \frac{3}{x ( ln ( x ) ) ^{3}} d x

What is the integral of (x^2)/(5-x^6) ?
The integral of (x^2)/(5-x^6) is 1/(6sqrt(5))(ln|(x^3)/(sqrt(5))+1|-ln|(x^3)/(sqrt(5))-1|)+C