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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3)/(x^2+3)

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

\frac{1}{2} (x^{2} + 3 - 3 ln x^{2} + 3) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 3}{2 u} d u

= \int \frac{u - 3}{2 u} d u

Take the constant out:

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

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

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

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

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= \frac{1}{2} (u - 3 ln ∣ u ∣)

Substitute back

u = x^{2} + 3

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

Add a constant to the solution

= \frac{1}{2} (x^{2} + 3 - 3 ln x^{2} + 3) + C

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

\int 9 (\frac{sin ( 2 x )}{sin ( x )}) d x

\int x sin (\frac{π x}{2}) d x

\int sin^{4} (t) cos (t) d t

\int 3 x (x^{2} + 4)^{2} d x

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