$$F(x)=\frac{d}{dx} \int_0^{e^x} log_e t\cdot cos^4 t\cdot dt$$
So I tried using Leibniz's rule, by differentiating under the integral
$$\frac{d}{d\alpha}F(x,\alpha)=\frac{d}{dx} \int_0^{e^x}\partial_{\alpha}\ log_e t\alpha\cdot cos^4 t\cdot dt$$
However, there were a few problems with finding the initial condition and all that.
Question is to evaluate the integral. This is a high school problem by the way. So please don't post any difficult answers (no offense). Hints would be nice too. And since I struggled with the initial condition (constant of integration) after using Leibniz (Feynman's differentiating under the integral), help with that part alone would be enough.
Best Answer
By the fundamental theorem of calculus You have $$\int_0^{e^x}\log_{e}(t)\cos^4(t)dt=G(e^x)-G(0)$$ where $G$ is an antiderivative of $g(t)=\log_{e}(t)\cos^4(t)$ and thus $$F(x)=\frac{d}{dx}\int_0^{e^x}\log_{e}(t)\cos^4(t)dt=\frac{d}{dx}G(e^x)=g(e^x)e^x$$ by the chain rule.