What is the integral of 10xln(x+5) ?

The integral of 10xln(x+5) is 10(ln(x+5)(1/2 (x+5)^2-5(x+5))-1/4 (x^2-10x-75))+C

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integral of 10xln(x+5)

\int 10 x ln (x + 5) d x

10 (ln (x + 5) (\frac{1}{2} (x + 5)^{2} - 5 (x + 5)) - \frac{1}{4} (x^{2} - 10 x - 75)) + C

Solution steps

\int 10 x ln (x + 5) d x

Take the constant out:

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

= 10 \cdot \int x ln (x + 5) d x

Apply u-substitution

: \int (u - 5) ln (u) d u

= 10 \cdot \int (u - 5) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u

= 10 (ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u)

\int \frac{1}{2} (u - 10) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

= 10 (ln (u) (\frac{u ^{2}}{2} - 5 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 10 u))

Substitute back

u = x + 5

= 10 (ln (x + 5) (\frac{( x + 5 ) ^{2}}{2} - 5 (x + 5)) - \frac{1}{2} (\frac{( x + 5 ) ^{2}}{2} - 10 (x + 5)))

Simplify

10 (ln (x + 5) (\frac{( x + 5 ) ^{2}}{2} - 5 (x + 5)) - \frac{1}{2} (\frac{( x + 5 ) ^{2}}{2} - 10 (x + 5))) : 10 (ln (x + 5) (\frac{1}{2} (x + 5)^{2} - 5 (x + 5)) - \frac{1}{4} (x^{2} - 10 x - 75))

= 10 (ln (x + 5) (\frac{1}{2} (x + 5)^{2} - 5 (x + 5)) - \frac{1}{4} (x^{2} - 10 x - 75))

Add a constant to the solution

= 10 (ln (x + 5) (\frac{1}{2} (x + 5)^{2} - 5 (x + 5)) - \frac{1}{4} (x^{2} - 10 x - 75)) + C

\int \frac{x}{8} d x

y^{^{'}} = y^{2}, y (0) = 1

y = x^{4} + 9 x^{2} - x, (1, 9)

e^{x + y} - y^{^{'}} = 0

\frac{d y}{d x} = (x - 2 y cot (2 x))

What is the integral of 10xln(x+5) ?
The integral of 10xln(x+5) is 10(ln(x+5)(1/2 (x+5)^2-5(x+5))-1/4 (x^2-10x-75))+C