What is the integral of ln(2+x) ?

The integral of ln(2+x) is 2ln(2+x)+xln(2+x)-2-x+C

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integral of ln(2+x)

integral

ln (2 + x)

2 ln (2 + x) + x ln (2 + x) - 2 - x + C

Solution steps

\int ln (2 + x) d x

Apply u-substitution

: \int ln (u) d u

= \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= u ln (u) - \int 1 d u

\int 1 d u = u

= u ln (u) - u

Substitute back

u = 2 + x

= (2 + x) ln (2 + x) - (2 + x)

Simplify

(2 + x) ln (2 + x) - (2 + x) : 2 ln (2 + x) + x ln (2 + x) - 2 - x

= 2 ln (2 + x) + x ln (2 + x) - 2 - x

Add a constant to the solution

= 2 ln (2 + x) + x ln (2 + x) - 2 - x + C

\frac{1}{x ^{2} + 5 [ 2 x ^{2} - 3 ]}

\frac{5 x \frac{2}{3} + 2}{3 x ^{2} + 5 x}

4 x^{2} - 62 x

(x^{2} + 1)^{0.5}

\frac{1}{( x + a )}

What is the integral of ln(2+x) ?
The integral of ln(2+x) is 2ln(2+x)+xln(2+x)-2-x+C