Mathematics

Integrate the following function :-
1) $$\log x$$
2) $${\cos ^{ - 1}}\left( {\sqrt x } \right)$$


SOLUTION
$$(i)\displaystyle\int \log x\,\,dx$$
$$=\displaystyle\int 1\times \log x\,\,dx$$
$$We\,\, ILATE$$
$$=\log x(\displaystyle\int 1.dx)-\int \dfrac{1}{x}\int (1-dx)dx$$
$$(\log x)\times x-\displaystyle\int \dfrac{1}{x}\times x\,\, dx$$
$$=x\log x-x+c$$

$$(ii) \displaystyle\int \cos ^{-1}\sqrt{x}\,\,dx$$
$$=\displaystyle\int \cos ^{-1}\sqrt{x}\times 1\times dx$$
          I                   II
$$=\cos^{-1}\sqrt{x}(\int1.dx)-\int \dfrac{1}{\sqrt{1-x^2}}(\int 1.dx)dx$$
$$(\cos^{-1}\sqrt{x})\times x+\int \dfrac{x}{\sqrt{1-x^2}}dx$$
let $$1-x^2=t^2$$
$$\therefore -2x\,dx=2t\,dt$$
$$\displaystyle I=x\cos^{-1}\sqrt{x}+\int \dfrac{-t\,dt}{t}$$
$$=x\cos^{-1}\sqrt{x}-t+c$$
$$=x\cos^{-1}\sqrt{x}-\sqrt{1-x^2}+c$$
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Subjective Medium Published on 17th 09, 2020
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