Question

\[\int\limits_{- 1}^1 5 x^4 \sqrt{x^5 + 1} dx\]

Answer 1

\[Let\ I = \int_{- 1}^1 5 x^4 \sqrt{x^5 + 1}\ d\ x . Then, \]
\[Let\ x^5 + 1 = t . Then, 5 x^4\ dx = dt\]
\[When\ x = - 1, t = 0\ and\ x = 1, t = 2\]
\[ \therefore I = \int_0^2 \sqrt{t}\ dt\]
\[ \Rightarrow I = \left[ \frac{2}{3} t^\frac{3}{2} \right]_0^2 \]
\[ \Rightarrow I = \frac{2}{3}\sqrt{8}\]
\[ \Rightarrow I = \frac{4\sqrt{2}}{3}\]