Q225 / 270 · NDA 2012 (Phase 1) · Mathematics

What is $\int\left(x^2+1\right)^{\frac{5}{2}} x d x$ equal to?

  • A
    $\left(x^2+1\right)^{\frac{7}{2}}+c$
  • B
    $\frac{2}{7}\left(x^2+1\right)^{\frac{7}{2}}+c$
  • Correct answer
    C
    $\frac{1}{7}\left(x^2+1\right)^{\frac{7}{2}}+c$
  • D
    None of the above

Solution

Put $x^2+1=t$ $\begin{aligned} & 2 x=d x=d t \\ & x d x=\frac{d t}{2} \quad \int(t)^{5 / 2} \frac{d t}{2}(C) \\ & \frac{1}{2} \frac{t^{7 / 2}}{7 / 2}+c=\frac{1}{7}\left(x^2+1\right)^{7 / 2}+c \end{aligned}$

Hence, the correct answer is option: 'C'.

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