What is the integral of ln(x)(5x+3) ?

The integral of ln(x)(5x+3) is 5/2 x^2ln(x)-5x^2+(5x^2)/4+(5x^2)/2+3xln(x)-3x+C

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integral of ln(x)(5x+3)

\int ln (x) (5 x + 3) d x

\frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C

Solution steps

\int ln (x) (5 x + 3) d x

Apply Integration By Parts

: (5 x + 3) (x ln (x) - x) - \int 5 (x ln (x) - x) d x

= (5 x + 3) (x ln (x) - x) - \int 5 (x ln (x) - x) d x

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

= (5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2})

Simplify

(5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2}) : \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

Add a constant to the solution

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C

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