What is the integral of xln(2x-3) ?

The integral of xln(2x-3) is 1/2 x^2ln(2x-3)-(x^2)/4-3/4 x-9/8 ln|2x-3|+C

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integral of xln(2x-3)

\int x ln (2 x - 3) d x

\frac{1}{2} x^{2} ln (2 x - 3) - \frac{x ^{2}}{4} - \frac{3}{4} x - \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

Solution steps

\int x ln (2 x - 3) d x

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{2 x - 3} d x = \frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣

= \frac{1}{2} x^{2} ln (2 x - 3) - (\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣)

Simplify

= \frac{1}{2} x^{2} ln (2 x - 3) - \frac{x ^{2}}{4} - \frac{3}{4} x - \frac{9}{8} ln ∣ 2 x - 3 ∣

Add a constant to the solution

= \frac{1}{2} x^{2} ln (2 x - 3) - \frac{x ^{2}}{4} - \frac{3}{4} x - \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

\int \frac{1}{x ^{2} + 10 x + 25} d x

n \to \infty lim (\frac{1}{n !})

x \to 0 lim (\frac{1}{x ^{2} ( x + 8 )})

y^{^{''}} + 4 y = sin (2 x)

\int_{0}^{8} 3 x d x

What is the integral of xln(2x-3) ?
The integral of xln(2x-3) is 1/2 x^2ln(2x-3)-(x^2)/4-3/4 x-9/8 ln|2x-3|+C