What is the integral of 10xln(x+2) ?

The integral of 10xln(x+2) is 10(ln(x+2)(1/2 (x+2)^2-2(x+2))-1/4 (x^2-4x-12))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 10xln(x+2)

\int 10 x ln (x + 2) d x

10 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12)) + C

Solution steps

\int 10 x ln (x + 2) 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 + 2) d x

Apply u-substitution

: \int (u - 2) ln (u) d u

= 10 \cdot \int (u - 2) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - 2 u) - \int \frac{1}{2} (u - 4) d u

= 10 (ln (u) (\frac{u ^{2}}{2} - 2 u) - \int \frac{1}{2} (u - 4) d u)

\int \frac{1}{2} (u - 4) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 4 u)

= 10 (ln (u) (\frac{u ^{2}}{2} - 2 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 4 u))

Substitute back

u = x + 2

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

Simplify

10 (ln (x + 2) (\frac{( x + 2 ) ^{2}}{2} - 2 (x + 2)) - \frac{1}{2} (\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2))) : 10 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12))

= 10 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12))

Add a constant to the solution

= 10 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12)) + C

\frac{\partial ^{2}}{\partial x \partial y} (x^{5} + x^{2} y + x + 3)

- 7

y^{^{''}} - 0.2 y^{^{'}} + 0.01 y = 0, y (0) = 10, y^{^{'}} (0) = b

\int cos (30 x) \cdot cos (4 x) \cdot cos (4 x) d x

y^{^{'}} = x + y, y (0) = 10

What is the integral of 10xln(x+2) ?
The integral of 10xln(x+2) is 10(ln(x+2)(1/2 (x+2)^2-2(x+2))-1/4 (x^2-4x-12))+C