Mathematics

Evaluate $$\displaystyle\int_{1}^{3}\dfrac{\cos (\log x)}{x}dx$$


SOLUTION
Let $$t=\log{x}\Rightarrow\,dt=\dfrac{1}{x}dx$$

When $$x=1\Rightarrow\,t=0$$

When $$x=3\Rightarrow\,t=\log{3}$$

$$\displaystyle\int_{1}^{3}{\dfrac{\cos{\left(\log{x}\right)}}{x}dx}$$

$$=\displaystyle\int_{0}^{\log{3}}{\cos{t}dt}$$

$$=\left[\sin{t}\right]_{0}^{\log{3}}$$

$$=\left[\sin{t}\right]_{0}^{\log{3}}$$

$$=\left[\sin{\log{3}}-\sin{0}\right]$$

$$=\sin{\log{3}}-0=\sin{\log{3}}$$

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Subjective Medium Published on 17th 09, 2020
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