For Chebyshev polynomials: \begin{align*} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^{2} - 1 \\ & \,\,\,\vdots \end{align*} is there a good way to approximate the following integral?: \begin{equation*} \int_{-1}^1 \exp\big\{a_0 T_0(x) + \cdots + a_k T_k(x)\big\} \; \text{d}x \end{equation*}
The motivation is that I thought it might be convenient to approximate the logarithm of a probability density function using Chebyshev polynomials, but I couldn't find anything about how to normalize the approximation.
One solution to your problem might be to write the exponential in terms of its power series. Then $\exp(P(x))$ boils down to computing $P(x)^s$ for $s = 1,2,\dots$.
Computing $P(x) \cdot P(x)$ can be done on the level of the Chebyshev coefficients $\alpha_i$ using multiplication matrices. The definition of these matrices can for instance be found in Section 3.1 of https://arxiv.org/pdf/1409.2789.pdf. This allows you to compute the Chebyshev coefficients of $P(x)^s$ for arbitrarily large $s$. You can then compute the integral for each term of the (truncated) power series individually.
You can truncate the power series as long as the maximum norm of $P(x)^s$ grows slower than $1/s!$, which should asymptotically apply for every polynomial.
Alternatively, it might also be worth to take a look at the exp function in the Chebfun package (https://www.chebfun.org/), which computes a Chebyshev polynomial approximation of the exponential of a Chebyshev polynomial.