SOLUTION: calculate the area under the curve y=f(x)= x^2 ln(x^3) from x=0 to x=1

Question 1031670: calculate the area under the curve y=f(x)= x^2 ln(x^3) from x=0 to x=1
Found 2 solutions by Alan3354, rothauserc:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
calculate the area under the curve y=f(x)= x^2 ln(x^3) from x=0 to x=1
---------
f(x) = 3x^2*ln(x)
INT(x) = 3*(x^3*ln(x)/3 - x^3/9) --- ignore the constant of integration
INT(1) = 3*1*0/3 - 3*1/9 = -1/3
INT(0) = 0
--> area = 1/3 sq unit, below the x-axis.

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
integrate x^2 ln(x^3) dx
:
substitute u = x^3 and du = 3 x^2 dx = 1/3 integral ln(u) du
:
For the integrand ln(u), integrate by parts, integral f dg = f g- integral g df, where f = ln(u), dg = du,
:
df = 1/u du, g = u
:
df = 1/3 u ln(u)-1/3 integral 1 du
:
The integral of 1 is u:
= 1/3 u ln(u)-u/3+constant
Substitute back for u = x^3:
= 1/3 x^3 ln(x^3)-x^3/3+constant
:
***********************************
y' = 1/3 x^3 (ln(x^3)-1)+constant
***********************************
:
we can evaluate the definite integral and forget about the constant
:
1/3(1)^3 ((ln(1^3)-1) - 0 =
:
***********************
(ln(1) / 3) - 1/3 = 1/3
***********************
:
note that area is positive
note that ln(1) = 0
:

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