What is the integral of xln^2(5x) ?

The integral of xln^2(5x) is 1/2 x^2ln^2(5x)-1/2 x^2ln(5x)+(x^2)/4+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of xln^2(5x)

\int x ln^{2} (5 x) d x

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

Solution steps

\int x ln^{2} (5 x) d x

Apply Integration By Parts

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

\int x ln (5 x) d x = \frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}

= \frac{1}{2} x^{2} ln^{2} (5 x) - (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4})

Simplify

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

Add a constant to the solution

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

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