$$ \left. \begin{array} { l }{ \int 16 \sqrt[ 5 ] { x ^ { 3 } } d x } \ { = [ ? ] \sqrt[ 5 ] { x ^ { \square } } + C } \end{array} ight. $$

Question

$$ \left. \begin{array} { l }{ \int 16 \sqrt[ 5 ] { x ^ { 3 } } d x } \ { = [ ? ] \sqrt[ 5 ] { x ^ { \square } } + C } \end{array} ight. $$

UpStudy AI · Accepted Answer

We start with the given integral:
$$
\int 16 \sqrt[5]{x^3} \, dx = \int 16 \, x^{\frac{3}{5}} \, dx.
$$

Factor out the constant:
$$
16 \int x^{\frac{3}{5}} \, dx.
$$

Apply the power rule for integration:
$$
\int x^a \, dx = \frac{x^{a+1}}{a+1} + C,
$$
with $$ a = \frac{3}{5} $$. Then:
$$
\int x^{\frac{3}{5}} \, dx = \frac{x^{\frac{3}{5} + 1}}{\frac{3}{5} + 1} + C.
$$

Calculate the new exponent:
$$
\frac{3}{5} + 1 = \frac{3}{5} + \frac{5}{5} = \frac{8}{5}.
$$

Thus, the integral becomes:
$$
\int x^{\frac{3}{5}} \, dx = \frac{x^{\frac{8}{5}}}{\frac{8}{5}} + C = \frac{5}{8} \, x^{\frac{8}{5}} + C.
$$

Now multiply by 16:
$$
16 \times \frac{5}{8} \, x^{\frac{8}{5}} = 10 \, x^{\frac{8}{5}}.
$$

Express $$ x^{\frac{8}{5}} $$ using radical notation:
$$
x^{\frac{8}{5}} = \sqrt[5]{x^8}.
$$

Therefore, the final answer is:
$$
\int 16 \sqrt[5]{x^3} \, dx = 10 \sqrt[5]{x^8} + C.
$$
$$ \int 16 \sqrt[5]{x^3} \, dx = 10 \sqrt[5]{x^8} + C $$

\( \left. \begin{array} { l }{ \int 16 \sqrt[ 5 ] { x ^ { 3 } } d x } \\ { = [ ? ] \sqrt[ 5 ] { x ^ { \square } } + C } \end{array} \right. \)

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