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If  \int \sqrt{\frac{cosx-cos^{3}x}{\left ( 1-cos^{3}x \right )}}dx = f (x) + c, then f (x) is equal to

  • Option 1)

    \frac{2}{3}\sin^{-1}\left ( cos^{3/2}x \right )

  • Option 2)

    \frac{3}{2}\sin^{-1}\left ( cos^{3/2}x \right )

  • Option 3)

    \frac{2}{3}\cos^{-1}\left ( cos^{3/2}x \right )

  • Option 4)

    none

 

Answers (1)

best_answer

As we learnt

Integration of Rational and irrational function -

Integration in the form of : 

(i) \int \frac{dx}{x^{2}+a^{2}} (ii) \int \frac{dx}{x^{2}-a^{2}} (iii) \int \frac{dx}{a^{2}-x^{2}}

(iv) \int \frac{dx}{\sqrt{x^{2}+a^{2}}} (v) \frac{dx}{\sqrt{x^{2}-a^{2}}} (vi) \frac{dx}{\sqrt{a^{2}-x^{2}}}

-

 

 Let I=\int {\sqrt {\frac{{\cos x - {{\cos }^3}x}}{{(1 - {{\cos }^3}x)}}} } d\,x=\int {\frac{{\sqrt {\cos \,x} \,(\sqrt {1 - {{\cos }^2}x} )}}{{\sqrt {1\, - {{\left( {{{\cos }^{\frac{3}{2}}}x} \right)}^2}} }}\,d\,x}

    =\int {\frac{{\sqrt {\cos \,x} \,(\sin x)}}{{\sqrt {1\, - {{\left( {{{\cos }^{3/2}}x} \right)}^2}} }}\,d\,x}

If\; \,{\cos ^{\frac{3}{2}}}\,\,x\, = \,p\,,\,then\,\,\,\left( { - \frac{3}{2}{{\cos }^{\frac{1}{2}}}x\,\sin \,x} \right)\,d\,x = dp\,\,

I= - \frac{2}{3}\int {\frac{{dp}}{{\sqrt {1 - {p^2}} }}} \,\,\,\,= - \,\frac{2}{3}\,{\sin ^{ - 1}}\,\left( {{{\cos }^{\frac{3}{2}}}x} \right)=\,\frac{2}{3}\,{\cos ^{ - 1}}\,\left( {{{\cos }^{\frac{3}{2}}}x} \right)+c_{1}


Option 1)

\frac{2}{3}\sin^{-1}\left ( cos^{3/2}x \right )

Option 2)

\frac{3}{2}\sin^{-1}\left ( cos^{3/2}x \right )

Option 3)

\frac{2}{3}\cos^{-1}\left ( cos^{3/2}x \right )

Option 4)

none

Posted by

gaurav

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