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
Questions 203525
Subjects 9
Chapters 126
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