https://file.aiquickdraw.com/mathbot/file/1726236907743e032yykm.jpg

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.

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These integrals range from trigonometric to logarithmic to exponential

Solve the following integrals:
1. ∫√(1 + x + x² + x³) dx
2. ∫(√(1 - tan(x)) / sin(x)) dx
3. ∫e^x ln(1 + √(1 + x²)) dx
4. ∫(sin(x) / (1 + sin(x) + sin(2x))) dx

Explore advanced calculus problems including solving complex integrals involving trigonometric, logarithmic, and exponential functions. Learn step-by-step solutions to challenging integrals suitable for undergraduate mathematics students.

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