4. \( \int \frac{(x+3) d x}{\sqrt{1+x^{2}}} \)

Calculate the integral \( \int \frac{(x+3) dx}{\sqrt{1+x^{2}}} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{\left(x+3\right)}{\sqrt{1+x^{2}}} dx\) - step1: Remove the parentheses: \(\int \frac{x+3}{\sqrt{1+x^{2}}} dx\) - step2: Rewrite the fraction: \(\int \left(\frac{x}{\sqrt{1+x^{2}}}+\frac{3}{\sqrt{1+x^{2}}}\right) dx\) - step3: Use properties of integrals: \(\int \frac{x}{\sqrt{1+x^{2}}} dx+\int \frac{3}{\sqrt{1+x^{2}}} dx\) - step4: Evaluate the integral: \(\left(1+x^{2}\right)^{\frac{1}{2}}+\int \frac{3}{\sqrt{1+x^{2}}} dx\) - step5: Evaluate the integral: \(\left(1+x^{2}\right)^{\frac{1}{2}}+3\ln{\left(x+\sqrt{1+x^{2}}\right)}\) - step6: Simplify: \(\sqrt{1+x^{2}}+3\ln{\left(x+\sqrt{1+x^{2}}\right)}\) - step7: Add the constant of integral C: \(\sqrt{1+x^{2}}+3\ln{\left(x+\sqrt{1+x^{2}}\right)} + C, C \in \mathbb{R}\) La integral de \( \frac{(x+3) dx}{\sqrt{1+x^{2}}} \) es \( \sqrt{1+x^{2}}+3\ln{(x+\sqrt{1+x^{2}})} + C \), donde \( C \) es una constante real.