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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{x ^{5}}{x ^{6} + 5} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 2 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = x^{6} + 5

= 2 \cdot \frac{1}{6} ln x^{6} + 5

Simplify

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

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

Add a constant to the solution

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

\int e^{3 + 3 y} d y

\int \frac{5}{6} d x

\int \frac{x ^{2}}{( x ^{2} - 9 ) ^{2}} d x

\int \frac{x}{( x - 1 ) ( 2 x + 5 ) ^{2} ( 3 x - 1 ) ^{2}} d x

\int \frac{5 x ^{2}}{( 1 - x ^{2} ) ^{\frac{5}{2}}} d x

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