Let the equation of the curve passing through the point \((0,1)\) be given by \(y=\int x^3 e^{x^4} d x\). If…
Let the equation of the curve passing through the point \((0,1)\) be given by \(y=\int x^3 e^{x^4} d x\). If the equation of the curve is written in the form \(x=f(y)\), then \(f(y)=\)
It is given that, \(y=\int x^3 e^{x^4} d x\)
Let \(x^4=t \Rightarrow 4 x^3 d x=d t\)
\(\therefore\) So, \(y=\frac{1}{4} \int e^t d t=\frac{e^t}{4}+C=\frac{e^{x^4}}{4}+C\)
\(\because\) The curve \(y=\frac{e^{x^4}}{4}+C\) passes through point \((0,1)\), so \(C=\frac{3}{4} \Rightarrow y=\frac{e^{x^4}}{4}+\frac{3}{4} \Rightarrow e^{x^4}=4 y-3\)
\(\Rightarrow \quad x^4=\log _e|4 y-3| \Rightarrow x=\left(\log _e|4 y-3|\right)^{1 / 4}\)
Hence, option (c) is correct.