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

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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

= \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

\int 2 x + 3 y d x

(2 y + 2 t - 3) d t + (8 y + 2 t - 1) d y = 0, y (- 1) = 2

\frac{d}{d x} (lo g_{8} (\frac{x ^{3} 8 ^{x}}{3 x + 1}))

((2 x + 1)^{- 1})^{^{'}}

f (x) = 2 x - \frac{1}{2 x}

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