Integral of udu

Dec 2009
200
35
Arkansas
\(\displaystyle \textrm{given}\,\,\,\int 3(5+3x)dx\)
\(\displaystyle \textrm{say that}\,\,\, u=(5+3x) \,\,\,\textrm{and}\,\,\, u'=\frac{d(3x)}{dx} \therefore\)
\(\displaystyle \int 3(5+3x)^6dx=\int u^6u'dx\)

\(\displaystyle \textrm{ This is were I get lost. It says that the answer to this would be}\,\,\,\frac{u^7}{7}+C\)

\(\displaystyle \textrm{But u' is a scalar so:} \,\,\, \int u^6u'dx=u'\int u^6dx\)
\(\displaystyle \textrm{and thus}\,\,\,\frac{d(3x)}{dx}\frac{u^7}{7}=3\frac{u^7}{7}\)

\(\displaystyle \textrm{So why does this book say}\,\, \frac{u^7}{7}\,\, \textrm{and not:} \,\, 3\frac{u^7}{7}\)

\(\displaystyle \textrm{thanks in advance}\) (Bow)
 
Dec 2009
3,120
1,342
\(\displaystyle \textrm{given}\,\,\,\int 3(5+3x)dx\)
\(\displaystyle \textrm{say that}\,\,\, u=(5+3x) \,\,\,\textrm{and}\,\,\, u'=\frac{d(3x)}{dx} \therefore\)
\(\displaystyle \int 3(5+3x)^6dx=\int u^6u'dx\)

\(\displaystyle \textrm{ This is were I get lost. It says that the answer to this would be}\,\,\,\frac{u^7}{7}+C\)

\(\displaystyle \textrm{But u' is a scalar so:} \,\,\, \int u^6u'dx=u'\int u^6dx\)
\(\displaystyle \textrm{and thus}\,\,\,\frac{d(3x)}{dx}\frac{u^7}{7}=3\frac{u^7}{7}\)

\(\displaystyle \textrm{So why does this book say}\,\, \frac{u^7}{7}\,\, \textrm{and not:} \,\, 3\frac{u^7}{7}\)

\(\displaystyle \textrm{thanks in advance}\) (Bow)
Hi integral,

\(\displaystyle u=5+3x\)

\(\displaystyle \frac{du}{dx}=3\)

\(\displaystyle du=3dx\)

Hence

\(\displaystyle \int{u^6}3dx=\int{u^6}du\)