Use integral tables to evaluate the integral.

Ask by Morgan Warren. in the United States

Jan 23,2025

Question

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

UpStudy AI · Accepted Answer

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:
  $$\int \left(\sqrt{169+x}ight)^{-1}x dx$$
- 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:
  $$\begin{align}&du=dx\&v=2\left(169+xight)^{\frac{1}{2}}\end{align}$$
- step5: Substitute the values into formula:
  $$x	imes 2\left(169+xight)^{\frac{1}{2}}-\int 1	imes 2\left(169+xight)^{\frac{1}{2}} dx$$
- step6: Calculate:
  $$2x\left(169+xight)^{\frac{1}{2}}-\int 2\left(169+xight)^{\frac{1}{2}} dx$$
- step7: Evaluate the integral:
  $$2x\left(169+xight)^{\frac{1}{2}}-\frac{4\left(169+xight)^{\frac{3}{2}}}{3}$$
- step8: Simplify:
  $$2x\sqrt{169+x}-\frac{4\left(169+xight)^{\frac{3}{2}}}{3}$$
- step9: Simplify:
  $$2x\sqrt{169+x}-\frac{4\sqrt{\left(169+xight)^{3}}}{3}$$
- step10: Expand the expression:
  $$2x\sqrt{169+x}-\frac{\left(676+4xight)\sqrt{169+x}}{3}$$
- step11: Add the constant of integral C:
  $$2x\sqrt{169+x}-\frac{\left(676+4xight)\sqrt{169+x}}{3} + C, C \in \mathbb{R}$$
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.
The integral $$ \int \frac{x}{\sqrt{169+x}} d x $$ equals $$ 2x\sqrt{169+x}-\frac{(676+4x)\sqrt{169+x}}{3} + C $$, where $$ C $$ is a constant.

Use integral tables to evaluate the integral.

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