$\displaystyle\int_{-1}^{1} 5x^4\sqrt{x^5+1}\,dx$ equals
A$\dfrac{4\sqrt{2}}{3}$
B$\dfrac{2\sqrt{2}}{3}$
C$\dfrac{4}{3}$
D$2\sqrt{2}$
Answer & Solution
Correct answer: A. $\dfrac{4\sqrt{2}}{3}$
1. Put $t=x^5+1$, so $dt=5x^4\,dx$.
2. Change limits: $x=-1\Rightarrow t=0$; $x=1\Rightarrow t=2$.
3. Integral $=\int_0^2 \sqrt{t}\,dt=\dfrac{2}{3}\left[t^{3/2}\right]_0^2=\dfrac{2}{3}\cdot 2\sqrt{2}$.
4. $=\dfrac{4\sqrt{2}}{3}$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.340_
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