1. ## Integral of udu

$\textrm{given}\,\,\,\int 3(5+3x)dx$
$\textrm{say that}\,\,\, u=(5+3x) \,\,\,\textrm{and}\,\,\, u'=\frac{d(3x)}{dx} \therefore$
$\int 3(5+3x)^6dx=\int u^6u'dx$

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

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

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

$\textrm{thanks in advance}$

2. Originally Posted by integral
$\textrm{given}\,\,\,\int 3(5+3x)dx$
$\textrm{say that}\,\,\, u=(5+3x) \,\,\,\textrm{and}\,\,\, u'=\frac{d(3x)}{dx} \therefore$
$\int 3(5+3x)^6dx=\int u^6u'dx$

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

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

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

$\textrm{thanks in advance}$
Hi integral,

$u=5+3x$

$\frac{du}{dx}=3$

$du=3dx$

Hence

$\int{u^6}3dx=\int{u^6}du$

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