# Integral of udu

#### integral

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

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

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

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

$$\displaystyle \textrm{thanks in advance}$$ (Bow)

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

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

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

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

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

$$\displaystyle u=5+3x$$

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

$$\displaystyle du=3dx$$

Hence

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