Evaluate the indefinite integral using the method of substitution: $\int \frac{x^2}{1 + x^6} \, dx$.

Evaluate the integral using integration by parts: $\int x \sin(2x) \, dx$.

Find the area of the region bounded by the parabola $y = 4 - x^2$ and the x-axis ($y = 0$).

Evaluate the following integrals using Integration by Parts (ILATE rule): (a) $\int x \cos x \, dx$ (b) $\int \ln x \, dx$ (c) $\int x^2 e^x \, dx$