If x^5e^-4x^3,dx=148e^-4x^3f(x)+C, where C is a constant of integration, then f(x) is equa…

Question

If $\displaystyle\int x^5e^{-4x^3}\,dx=\dfrac1{48}e^{-4x^3}f(x)+C$, where C is a constant of integration, then $f(x)$ is equal to:

Accepted Answer

$I=\displaystyle\int x^5e^{-4x^3}dx$. Put $-4x^3=\theta\Rightarrow-12x^2dx=d\theta\Rightarrow x^2dx=-\dfrac{d\theta}{12}$, and $x^3=-\theta/4$:
$$I=\int\frac1{48}\theta e^{\theta}d\theta=\frac1{48}\left[\theta e^\theta-e^\theta\right]+C=\frac1{48}e^{-4x^3}(-4x^3-1)+C$$

So $f(x)=-4x^3-1$

Solutions · 1