What is the integral of xln(sqrt(5)x) ?

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

\int x ln (5 x) d x

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

Solution steps

\int x ln (5 x) d x

Expand

x ln (5 x) : x ln (x) + \frac{ln ( 5 ) x}{2}

= \int x ln (x) + \frac{ln ( 5 ) x}{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x ln (x) d x + \int \frac{ln ( 5 ) x}{2} d x

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

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

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

Add a constant to the solution

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

\int \frac{1}{x ( x + 1 )} d x

\int \frac{1}{( x ^{2} + 81 ) ^{\frac{3}{2}}} d x

x \to 8 lim (\frac{( x ^{2} - 5 x - 24 )}{x - 8})

\int (x + a)^{2} d x

7 + cot (x) - 2 csc (x)

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