What is the integral of 5xln(3-x) ?

The integral of 5xln(3-x) is 5(1/2 x^2ln(3-x)+1/2 (-(x^2)/2-3x-9ln|-x+3|))+C

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integral of 5xln(3-x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

= 5 (\frac{1}{2} x^{2} ln (3 - x) - \int - \frac{x ^{2}}{2 ( 3 - x )} d x)

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

= 5 (\frac{1}{2} x^{2} ln (3 - x) - (- \frac{1}{2} (- \frac{x ^{2}}{2} - 3 x - 9 ln ∣ - x + 3 ∣)))

Simplify

= 5 (\frac{1}{2} x^{2} ln (3 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 3 x - 9 ln ∣ - x + 3 ∣))

Add a constant to the solution

= 5 (\frac{1}{2} x^{2} ln (3 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 3 x - 9 ln ∣ - x + 3 ∣)) + C

y = (3 x + 1)^{2}, (0, 1)

f (x) = x^{5}

f (x) = \frac{1}{x ^{4}}, (- 2, \frac{1}{16})

\frac{d}{d x} (\frac{c ^{2} - x ^{2}}{2 x})

\int π (\frac{1}{1 + x ^{2}})^{2} d x

What is the integral of 5xln(3-x) ?
The integral of 5xln(3-x) is 5(1/2 x^2ln(3-x)+1/2 (-(x^2)/2-3x-9ln|-x+3|))+C