
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
Compute the following definite integrals:

(1) $\int_{0}^{2}{x(x+1)dx}$
(2) $\int_{1}^{2}{\left( {{e}^{2x}}+\dfrac{1}{x} \right)dx}$
(3) $\int_{0}^{\frac{\pi }{2}}{{{\sin }^{2}}\dfrac{x}{2}dx}$
(4) $\int_{0}^{2}{\sqrt{4-{{x}^{2}}}dx}$
(5) $\int_{0}^{\frac{\pi }{2}}{\dfrac{\cos 2x}{\cos x+\sin x}dx}$
(6) $\int_{-\frac{\pi }{4}}^{\frac{\pi }{4}}{(\cos x+\dfrac{1}{4}{{x}^{3}}+1)dx}$

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