Solutions - 0050-07

(1) Take out constants and insert given values:

$${\int }_{x_1}^{x_2}\frac{k\lambda h}{{(x^2+h^2)}^{\frac{3}{2}}}dx\gg \gg k\lambda h{\int }_{-15}^{15}\frac{dx}{{(x^2+9)}^{\frac{3}{2}}}$$

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(2) Notice $x^2+9$ pattern, so we should make use of the identity ${\tan }^2\theta +1={\sec }^2\theta$.

Select $x=3\tan \theta$ and thus $dx=3{\sec }^2\theta d\theta$. Then:

$${(x^2+9)}^{3/2}\gg \gg {(9{\sec }^2\theta )}^{3/2}\gg \gg 27{\sec }^3\theta$$

Adjust bounds:

$$\begin{array}{cc}x& =-15\gg \gg \theta ={\tan }^{-1}(-5)\\ x& =15\gg \gg \theta ={\tan }^{-1}(5)\end{array}$$

Then:

$$\begin{array}{c}\gg \gg k\lambda h{\int }_{{\tan }^{-1}(-5)}^{{\tan }^{-1}(5)}\frac{3{\sec }^2\theta }{27{\sec }^3\theta }d\theta \\ \\ \gg \gg \frac{k\lambda h}{9}{\int }_{{\tan }^{-1}(-5)}^{{\tan }^{-1}(5)}\cos \theta d\theta \end{array}$$

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(3) Integrate:

$$\begin{array}{c}\gg \gg {\frac{k\lambda h}{9}\sin \theta |}_{{\tan }^{-1}(-5)}^{{\tan }^{-1}(5)}\\ \\ \gg \gg \frac{k\lambda h}{9}\left(\sin ({\tan }^{-1}(5))-\sin ({\tan }^{-1}(-5))\right)\\ \\ \gg \gg \frac{k\lambda h}{9}\left(\sin ({\tan }^{-1}(5))+\sin ({\tan }^{-1}(5))\right)\\ \\ \gg \gg \frac{(6\times {10}^{-4})(8.99\times {10}^9)}{3}2\sin ({\tan }^{-1}(5))\end{array}$$

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(4) Compute $\sin ({\tan }^{-1}(5))$:

Draw a triangle expressing $\tan \theta ={\displaystyle \frac{5}{1}}$:

Therefore $\sin ({\tan }^{-1}(5)={\displaystyle \frac{5}{\sqrt{26}}}$. Then:

$$\gg \gg \frac{(6\times {10}^{-3})(8.99\times {10}^{10})}{3\sqrt{26}}$$