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

The integral of 2x(ln(x))^3 is x^2ln^3(x)-3/2 x^2ln^2(x)+3/2 x^2ln(x)-3/4 x^2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2x(ln(x))^3

\int 2 x (ln (x))^{3} d x

x^{2} ln^{3} (x) - \frac{3}{2} x^{2} ln^{2} (x) + \frac{3}{2} x^{2} ln (x) - \frac{3}{4} x^{2} + C

Solution steps

\int 2 x (ln (x))^{3} d x

Take the constant out:

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

= 2 \cdot \int x (ln (x))^{3} d x

Simplify

= 2 \cdot \int x ln^{3} (x) d x

Apply Integration By Parts

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

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

\int \frac{3}{2} x ln^{2} (x) d x = \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})

= 2 (\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}))

Simplify

2 (\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})) : x^{2} ln^{3} (x) - \frac{3}{2} x^{2} ln^{2} (x) + \frac{3}{2} x^{2} ln (x) - \frac{3}{4} x^{2}

= x^{2} ln^{3} (x) - \frac{3}{2} x^{2} ln^{2} (x) + \frac{3}{2} x^{2} ln (x) - \frac{3}{4} x^{2}

Add a constant to the solution

= x^{2} ln^{3} (x) - \frac{3}{2} x^{2} ln^{2} (x) + \frac{3}{2} x^{2} ln (x) - \frac{3}{4} x^{2} + C

\int e^{3 z} cos (3 z) d z

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

\int \frac{1}{4 + 81 x ^{2}} d x

\int (3 + 2 x y) d x

\int e^{x} - cos (x) + 6 x + 4 d x

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