$$\frac{\mathrm{d}}{\mathrm{d}x} \int_{2x}^{3x+1}\! \sin\left(t^4\right)\, \mathrm{d}t$$
could you just use the Fundamental Theorem of Calculus to get
$$\sin\left(t^4\right) \bigg|_{2x}^{3x+1}$$
ie
$$\frac{\mathrm{d}}{\mathrm{d}x} \int_{2x}^{3x+1} \sin\left(t^4\right)\mathrm{d}t \space =\space \sin\left((3x+1)^4\right)-\sin\left((2x)^4\right)$$
$\endgroup$ 22 Answers
$\begingroup$Let $F(t)$ be an antiderivative of $\sin(t^4)$. Then what you have is $$ \frac{d}{dx}(F(3x+1)-F(2x)). $$ You can use the chain rule and the fact that $F'(x)=\sin(x^4)$ to get what you want.
$\endgroup$ $\begingroup$In the most general manner, the fundamental theorem of calculus gives $$\frac{d}{dx} \int_{a(x)}^{b(x)} F (t) \, dt=F \Big(b(x) \Big) b'(x)-F \Big(a(x) \Big) a'(x)$$
$\endgroup$
