4. \( \int\left(x^{2}+3\right)^{2} d x \)

To solve the integral \( \int\left(x^{2}+3\right)^{2} d x \), we can start by expanding the integrand: \[ \left(x^{2}+3\right)^{2} = x^{4} + 6x^{2} + 9 \] Now, we can integrate term by term: \[ \int (x^{4} + 6x^{2} + 9) \, dx = \int x^{4} \, dx + 6 \int x^{2} \, dx + 9 \int 1 \, dx \] Calculating each integral: \[ \int x^{4} \, dx = \frac{x^{5}}{5} + C, \] \[ 6 \int x^{2} \, dx = 6 \cdot \frac{x^{3}}{3} = 2x^{3}, \] \[ 9 \int 1 \, dx = 9x. \] Putting it all together, we have: \[ \int(x^{2}+3)^{2} dx = \frac{x^{5}}{5} + 2x^{3} + 9x + C, \] where \( C \) is the constant of integration.