Evaluate the definite integrals`int_0^(pi/4)(2sec^2x+x^3+2)dx`

Explanation

To evaluate the definite integral \( \int_0^{\frac{\pi}{4}} (2 \sec^2 x + x^3 + 2) \, dx \), we will break it down into simpler parts and integrate each term separately.

### Step 1: Split the Integral
We can split the integral into three separate integrals:
\[ \int_0^{\frac{\pi}{4}} (2 \sec^2 x + x^3 + 2) \, dx = \int_0^{\frac{\pi}{4}} 2 \sec^2 x \, dx + \int_0^{\frac{\pi}{4}} x^3 \, dx + \int_0^{\frac{\pi}{4}} 2 \, dx \]

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