document.write( "Question 1031670: calculate the area under the curve y=f(x)= x^2 ln(x^3) from x=0 to x=1 \n" ); document.write( "
Algebra.Com's Answer #646360 by rothauserc(4718)\"\" \"About 
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integrate x^2 ln(x^3) dx
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\n" ); document.write( "substitute u = x^3 and du = 3 x^2 dx = 1/3 integral ln(u) du
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\n" ); document.write( "For the integrand ln(u), integrate by parts, integral f dg = f g- integral g df, where f = ln(u), dg = du,
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\n" ); document.write( "df = 1/u du, g = u
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\n" ); document.write( "df = 1/3 u ln(u)-1/3 integral 1 du
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\n" ); document.write( "The integral of 1 is u:
\n" ); document.write( " = 1/3 u ln(u)-u/3+constant
\n" ); document.write( "Substitute back for u = x^3:
\n" ); document.write( " = 1/3 x^3 ln(x^3)-x^3/3+constant
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\n" ); document.write( "y' = 1/3 x^3 (ln(x^3)-1)+constant
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\n" ); document.write( "we can evaluate the definite integral and forget about the constant
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\n" ); document.write( "1/3(1)^3 ((ln(1^3)-1) - 0 =
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\n" ); document.write( "(ln(1) / 3) - 1/3 = 1/3
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\n" ); document.write( "note that area is positive
\n" ); document.write( "note that ln(1) = 0
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