That title is a mouthful.
Say $\theta$ is a function of t such that $\theta(t)$, and I want to integrate:
$$\int\sin\theta dt$$
In this case I probably can't just treat $\sin\theta$ as a constant and take it out the integral…
Integrate a function with respect to a variable that that function is itself also a function of
integration
Best Answer
Assuming that there is a function $g$ such that $\theta=g(t)$ for all $t$, we can write your integral as $$ I=\int \sin\left(g(t)\right) \, dt \, . $$ The issue is that there is no general formula for the value of this integral. For instance, if $g(t)=e^t$, then I believe that $I$ cannot be expressed in terms of elementary functions. However, if the function $\sin \circ g$ is continuous on $\mathbb{R}$, then $$ I=\int_{0}^{t}\sin(g(x)) \, dx+C $$ This follows from the first fundamental theorem of calculus.