FORM GR1268 Question 32

Question:
\(\displaystyle\frac{d}{dx}\int_{x^3}^{x^4}d^{t^2}dt=\)

(A) \(\displaystyle e^{x^6}\left(e^{x^8-x^6}-1\right)\)

(B) \(\displaystyle 4x^3e^{x^8}\)

(C) \(\displaystyle \frac{1}{\sqrt{1-e^{x^2}}}\)

(D) \(\displaystyle \frac{e^{x^2}}{x^2}-1\)

(E) \(\displaystyle x^2e^{x^6}\left(4xe^{x^8-x^6}-3\right)\)

Answer:
(E)

Answer Key:
Let \(f(t)=e^{t^2}\) and \(F'(t)=f(t)\).
\[\begin{align}
\frac{d}{dx}\int_{x^3}^{x^4}f(t)dt
&= \frac{d}{dx}\left.F(t)\right|_{x^3}^{x^4}\\
&=\frac{d}{dx}\left(F(x^4)-F(x^3)\right)\\
&=\frac{d}{dx}F(x^4)-\frac{d}{dx}F(x^3)\\
&=\frac{d}{dx^4}F(x^4)\cdot\frac{d}{dx}x^4-\frac{d}{dx^3}F(x^3)\cdot\frac{d}{dx}x^3\\
&=f(x^4)4x^3-f(x^3)3x^2\\
&=4x^3e^{(x^4)^2}-3x^2e^{(x^3)^2}\\
&=4x^3e^{x^8}-3x^2e^{x^6}\\
&=x^2e^{x^6}\left(4xe^{x^8-x^6}-3\right).
\end{align}\]