CUET · MATHS · PYQ PAPER 2025
The value of \(\int \frac{x^5}{\sqrt{1+x^3}}\)dx is
- A \(\frac{2}{9}\left(1+x^3\right)^{3 / 2}-\frac{2}{3}\left(1+x^3\right)^{1 / 2}+c\) : c is an arbitrary constant
- B \(\frac{2}{3}\left(1+x^3\right)^{3 / 2}-\frac{2}{3}\left(1+x^3\right)^{1 / 2}+c\) : c is an arbitrary constant
- C \(\frac{1}{3}\left(1+x^3\right)^{3 / 2}+\frac{1}{3}\left(1+x^3\right)^{1 / 2}+c\) : c is an arbitrary constant
- D \(\frac{2}{9}\left(1+x^3\right)^{3 / 2}+\frac{2}{3}\left(1+x^3\right)^{1 / 2}+c\) : c is an arbitrary constant
Answer & Solution
Correct Answer
(A) \(\frac{2}{9}\left(1+x^3\right)^{3 / 2}-\frac{2}{3}\left(1+x^3\right)^{1 / 2}+c\) : c is an arbitrary constant
Step-by-step Solution
Detailed explanation
Let \(u = 1+x^3 \Rightarrow du = 3x^2 dx \Rightarrow x^2 dx = \frac{1}{3}du\) and \(x^3 = u-1\). \(\int \frac{x^3 \cdot x^2}{\sqrt{1+x^3}}dx = \int \frac{(u-1)}{\sqrt{u}} \frac{1}{3}du\) \(= \frac{1}{3} \int (u^{1/2} - u^{-1/2})du\)…
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