Questions: Please show ALL work to earn credit. Evaluate the indefinite integral. [ int x^5 sqrt15+x^6 d x ]

Solution Steps

To evaluate the integral

\[ \int x^{5} \sqrt{15+x^{6}} \, dx, \]

we use the substitution \( u = 15 + x^{6} \). Then, the differential \( du \) is given by

\[ du = 6x^{5} \, dx \quad \Rightarrow \quad dx = \frac{du}{6x^{5}}. \]

Substituting \( u \) into the integral, we have:

\[ \int x^{5} \sqrt{15+x^{6}} \, dx = \int x^{5} \sqrt{u} \cdot \frac{du}{6x^{5}} = \frac{1}{6} \int \sqrt{u} \, du. \]

The integral of \( \sqrt{u} \) is

\[ \int \sqrt{u} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}. \]

Thus, we have:

\[ \frac{1}{6} \cdot \frac{2}{3} u^{3/2} = \frac{1}{9} u^{3/2}. \]

Now, substituting back \( u = 15 + x^{6} \), we get:

\[ \frac{1}{9} (15 + x^{6})^{3/2}. \]

Final Answer