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Previous Year Question (PYQs)

Let the equation of a curve passing through $(0,1)$ be $y=\displaystyle\int x^{2}e^{x^{3}}\,dx$. If the curve is written as $x=f(y)$, then $f(y)$ is –




Solution

Solution: Let $t=x^3 \Rightarrow dt=3x^2dx$, so $y=\dfrac{1}{3}e^{x^3}+C$. Using $(0,1)$: $1=\dfrac{1}{3}+C \Rightarrow C=\dfrac{2}{3}$. Hence $3y-2=e^{x^3} \Rightarrow x^3=\log_e(3y-2)$, so $x=\sqrt[3]{\log_e(3y-2)}$.

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