What is the integral of (2x^3)/(x^4+5) ?

The integral of (2x^3)/(x^4+5) is 1/2 ln|x^4+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^3)/(x^4+5)

\int \frac{2 x ^{3}}{x ^{4} + 5} d x

\frac{1}{2} ln x^{4} + 5 + C

Solution steps

\int \frac{2 x ^{3}}{x ^{4} + 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 ^{3}}{x ^{4} + 5} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= 2 \cdot \int \frac{1}{4 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}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 5

= 2 \cdot \frac{1}{4} ln x^{4} + 5

Simplify

2 \cdot \frac{1}{4} ln x^{4} + 5 : \frac{1}{2} ln x^{4} + 5

= \frac{1}{2} ln x^{4} + 5

Add a constant to the solution

= \frac{1}{2} ln x^{4} + 5 + C

a cos^{2} (x) + b x

\int (sin (x) - 4 cos (x)) d x

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

\frac{d}{d x} (ln (3 x + 4))

x \to 6 + lim (x 6 - x)

What is the integral of (2x^3)/(x^4+5) ?
The integral of (2x^3)/(x^4+5) is 1/2 ln|x^4+5|+C