Integration problem and fundamental theorem of calculus 1

Question

$$\frac{\mathrm d}{\mathrm dx}\int_1^x \mathrm e^{\cos t}\,\mathrm dt$$ I get the solution $\mathrm e^{\cos x}$ however it is not the exact solution. The correct solution $\mathrm e^{\cos t}$.

Answer 1

I think that your solution is correct; $e^{\cos t}$ given as a solution must be a mistake in the book. The variable $t$ is a dummy variable that only has meaning within the integral.

Answer 2

d/d∫e^cosd =d/dx[e^cos/sint] for 1,x

d/d∫e^cosd = d/dx[e^cosx/sinx-e^cos1/sin1] = -sinxe^cosx/sinx= - e^cosx

so it is probably a mistake from you book first you have to integrate and then differentiate in order to get the result