Let's break down each of the integrals presented in your image one by one:

1. $\int \sqrt{1 + x + x^2 + x^3} \, dx = ?$

This integral involves a polynomial under the square root. It looks complicated, but we could attempt substitution or simplification by factorization. This would take a more detailed approach, such as applying integration techniques like trigonometric substitution or series expansion.

2. $\int \frac{\sqrt{1 - \tan(x)}}{\sin(x)} \, dx = ?$

This one has a square root involving a trigonometric function. A substitution like $u = \tan(x)$ or some trigonometric identities might simplify the expression, although the square root adds complexity.

3. $\int e^x \ln\left(1 + \sqrt{1 + x^2}\right) \, dx = ?$

This is a more advanced integral involving both $e^x$ and a logarithmic function. For this type, integration by parts might be useful, as well as considering expansions or simplifications within the logarithmic term.

4. $\int \frac{\sin(x)}{1 + \sin(x) + \sin(2x)} \, dx = ?$

This involves trigonometric identities. We can simplify $\sin(2x)$ as $2\sin(x)\cos(x)$, which may lead to a more manageable expression. Substitution or recognizing a pattern in the denominator would help here.

These integrals range from trigonometric to logarithmic to exponential