Use integral tables to evaluate the integral. \( \int \frac{x}{\sqrt{169+x}} d x \)

Calculate the integral \( \int \frac{x}{\sqrt{169+x}} d x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int \frac{x}{\sqrt{169+x}} dx\) - step1: Evaluate the integral: \(\int \left(\sqrt{169+x}\right)^{-1}x dx\) - step2: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=\frac{1}{\sqrt{169+x}}dx\end{align}\) - step3: Calculate the derivative: \(\begin{align}&du=dx\\&dv=\frac{1}{\sqrt{169+x}}dx\end{align}\) - step4: Evaluate the integral: \(\begin{align}&du=dx\\&v=2\left(169+x\right)^{\frac{1}{2}}\end{align}\) - step5: Substitute the values into formula: \(x\times 2\left(169+x\right)^{\frac{1}{2}}-\int 1\times 2\left(169+x\right)^{\frac{1}{2}} dx\) - step6: Calculate: \(2x\left(169+x\right)^{\frac{1}{2}}-\int 2\left(169+x\right)^{\frac{1}{2}} dx\) - step7: Evaluate the integral: \(2x\left(169+x\right)^{\frac{1}{2}}-\frac{4\left(169+x\right)^{\frac{3}{2}}}{3}\) - step8: Simplify: \(2x\sqrt{169+x}-\frac{4\left(169+x\right)^{\frac{3}{2}}}{3}\) - step9: Simplify: \(2x\sqrt{169+x}-\frac{4\sqrt{\left(169+x\right)^{3}}}{3}\) - step10: Expand the expression: \(2x\sqrt{169+x}-\frac{\left(676+4x\right)\sqrt{169+x}}{3}\) - step11: Add the constant of integral C: \(2x\sqrt{169+x}-\frac{\left(676+4x\right)\sqrt{169+x}}{3} + C, C \in \mathbb{R}\) The integral \( \int \frac{x}{\sqrt{169+x}} d x \) evaluates to \( 2x\sqrt{169+x}-\frac{(676+4x)\sqrt{169+x}}{3} + C \), where \( C \) is an arbitrary constant.