Question

Let the equation of a curve passing through the point $(0,1)$ be given by $y=\int x^2.e^{x^3}dx.$ If the equation of the curve is written in the form $x=f\left(y\right)$ then $f\left(y\right)$ is

• A. $\sqrt{{\log }_e(3y-2)}$
• B. $\sqrt[3]{3{\log }_e(3y-2)}$
• C. $\sqrt[3]{3{\log }_e(2-3y)}$
• D. none of these

Answer 1

The correct option is B $\sqrt[3]{3{\log }_e(3y-2)}$

Substitute $x^3=t$

$\Rightarrow 3x^{2}dx=dt$

The equation becomes $y=\frac{1}{3}\int e^{t}dt=\frac{1}{3}{e}^t+c=\frac{1}{3}{e}^{x^3}+c$

Given that the curve passes through $(0,1)$. Using this we get

$c=\frac{2}{3}$

So, equation becomes $3y=e^{x^3}+2$

$\Rightarrow e^{x^3}=3y-2$

$\Rightarrow x^3=\ln (3y-2)$

$\Rightarrow x=\sqrt[3]{3\ln (3y-2)}$