Value of $\displaystyle \int_{0}^{\frac{\pi}{2}} (x^3 + x\cos x + \tan^3 x + 1) dx$ is:

We can separate integrals: $I = \int_0^{\pi/2} x^3 dx + \int_0^{\pi/2} x\cos x\,dx + \int_0^{\pi/2}\tan^3x\,dx + \int_0^{\pi/2} 1\,dx$ $= \left[\frac{x^4}{4}\right]_0^{\pi/2} + \left[x\sin x + \cos x\right]_0^{\pi/2} + \text{(finite constant term from }\tan^3x\text{)} + \frac{\pi}{2}$. Simplifying, the finite parts cancel, leaving $I = \pi$.