Take the integral:
int x log(x + 1) dx∫xlog(x+1)dx
For the integrand x ln(x + 1)xln(x+1), substitute u = x + 1u=x+1 and du = dxdu=dx:
= int(u - 1) ln(u) du=∫(u−1)ln(u)du
For the integrand (u - 1) log(u)(u−1)log(u), integrate by parts, int f dg = f g - int g df∫fdg=fg−∫gdf where
f = ln(u)f=ln(u), dg = (u - 1) dudg=(u−1)du, df = 1/u dudf=1udu, g = 1/2 (u - 1)^2g=12(u−1)2:
= (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 int(u - 1)^2/u du=ln(u)2−uln(u)+12u2ln(u)−12∫(u−1)2udu
For the integrand (u - 1)^2/u(u−1)2u, substitute s = u - 1s=u−1 and ds = duds=du:
= (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 int s^2/(s + 1) ds=ln(u)2−uln(u)+12u2ln(u)−12∫s2s+1ds
For the integrand s^2/(s + 1)s2s+1, do long division:
= (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 integral(s + 1/(s + 1) - 1) ds=ln(u)2−uln(u)+12u2ln(u)−12∫egral(s+1s+1−1)ds
Integrate the sum term by term and factor out constants:
= (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 int/(s + 1) ds - 1/2 int s ds + 1/2 int ds=ln(u)2−uln(u)+12u2ln(u)−12∫s+1ds−12∫sds+12∫ds
For the integrand 1/(s + 1)1s+1, substitute p = s + 1p=s+1 and dp = dsdp=ds:
= (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 int1/p dp - 1/2 int s ds + 1/2 int1 ds=ln(u)2−uln(u)+12u2ln(u)−12∫1pdp−12∫sds+12∫1ds
The integral of 1/p is log(p)1pislog(p):
= -(ln(p))/2 + (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) - 1/2 int s ds + 1/2 int1 ds=−ln(p)2+ln(u)2−uln(u)+12u2ln(u)−12∫sds+12∫1ds
The integral of ss is s^2/2s22:
= -s^2/4 - (ln(p))/2 + (ln(u))/2 - u ln(u) + 1/2 u^2 ln(u) + 1/2 int1 ds=−s24−ln(p)2+ln(u)2−uln(u)+12u2ln(u)+12∫1ds
The integral of 11 is ss:
= -(ln(p))/2 - s^2/4 + s/2 + 1/2 u^2 ln(u) - u ln(u) + (ln(u))/2 + "constant"=−ln(p)2−s24+s2+12u2ln(u)−uln(u)+ln(u)2+constant
Substitute back for p = s + 1p=s+1:
= -s^2/4 + s/2 - 1/2 ln(s + 1) + 1/2 u^2 ln(u) - u ln(u) + (ln(u))/2 + "constant"=−s24+s2−12ln(s+1)+12u2ln(u)−uln(u)+ln(u)2+constant
Substitute back for s = u - 1s=u−1:
= 1/2 u^2 ln(u) - 1/4 (u - 1)^2 + (u - 1)/2 - u ln(u) + "constant"=12u2ln(u)−14(u−1)2+u−12−uln(u)+constant
Substitute back for u = x + 1u=x+1:
= -x^2/4 + x/2 + 1/2 (x + 1)^2 ln(x + 1) - (x + 1) ln(x + 1) + "constant"=−x24+x2+12(x+1)2ln(x+1)−(x+1)ln(x+1)+constant
Which is equal to:
Answer:
1/4 (2 (x^2 - 1) ln(x + 1) - (x - 2) x) + "constant"14(2(x2−1)ln(x+1)−(x−2)x)+constant
