Evaluate $\displaystyle \int_{-1}^{1} 5x^4 \sqrt{x^5 + 1} \, dx$.

$\frac{4\sqrt{2}}{3}$

The correct answer is Option (2) → $\frac{4\sqrt{2}}{3}$

Put $t = x^5 + 1$, then $dt = 5x^4 \, dx$.

Therefore, $\int 5x^4 \sqrt{x^5 + 1} \, dx = \int \sqrt{t} \, dt = \frac{2}{3} t^{\frac{3}{2}} = \frac{2}{3} (x^5 + 1)^{\frac{3}{2}}$

Hence, $\int_{-1}^{1} 5x^4 \sqrt{x^5 + 1} \, dx = \frac{2}{3} \left[ (x^5 + 1)^{\frac{3}{2}} \right]_{-1}^{1}$

$= \frac{2}{3} \left[ (1^5 + 1)^{\frac{3}{2}} - ((-1)^5 + 1)^{\frac{3}{2}} \right]$

$= \frac{2}{3} \left[ 2^{\frac{3}{2}} - 0^{\frac{3}{2}} \right] = \frac{2}{3} (2\sqrt{2}) = \frac{4\sqrt{2}}{3}$