I'm trying to integrate $\int_{\mathbf{x}}{\mathbf{F}\cdot{d\mathbf{s}}}$ where $$\mathbf{F}=x^2\mathbf{i}+\cos{y}\sin{z}\mathbf{j}+\sin{y}\cos{z}\mathbf{k}$$and $$\mathbf{x}:[0,1]\rightarrow\mathbb{R}^3; \mathbf{x}(t)=(t^2+1,e^t,e^{2t})$$ I'm at the point where I have $$\int_{\mathbf{x}}{\mathbf{F}\cdot{d\mathbf{s}}}=\int_{\mathbf{x}}{x^2dx+\cos{y}\sin{z}dy+\sin{y}\cos{z}dz}$$$$=\int_0^1{(t^2+1)^22tdt+\int_0^1e^t[\cos{e^t}\sin{e^{2t}}+2e^t\cos{e^{2t}}\sin{e^t}]dt}$$ Thought I could use the $\sin{(A+B)}$ rule but the $2e^t$ is in front of the second term. The second thought was substitution, but I can't justify both $e^t$ and $e^{2t}$ terms. What am I missing?
EDIT: corrected typo in vector $\mathbf{x}(t)$. I should also point out that I'm not worried about the integral on the left. That poses no issues....
You are missing the opportunity to use the fundamental theorem of calculus for line integrals. As the title says, the field is conservative: it is the gradient of $x^3/3+\sin y\sin z$. Evaluate the potential function at the endpoints, and you are done.