
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider the Newton-Leibniz formula. If the function \( f(x) \) is continuous on the interval \([a, b]\), then

$$
\int_{a}^{b} f(x) d x=\left.F(x)\right|_{a} ^{b}=F(b)-F(a),
$$

where \( F(x) \) is the antiderivative of \( f(x) \), i.e., \( F^{\prime}(x)=f(x) \). Compute the integrals:
1) \(\int_{0}^{8}(\sqrt{2 x}+\sqrt[3]{x}) d x\)
2) \(\int_{0}^{1} \sqrt{1+t} d t\)
3) \(\int_{2}^{3} \frac{d x}{\sqrt{5+4 x-x^{2}}}\)
4) \(\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec ^{2} \alpha d \alpha\)

Present the answer in LaTex format: \boxed{Your answer}
user