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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate: $\int \limits_{0}^{\large\frac{\pi}{2}} \large\frac{\cos x}{(1+\sin x)(2+ \sin x)}$$ dx $

\[\begin {array} {1 1} (a)\;\log \frac{4}{3} \\ (b)\;\log \frac{1}{3} \\ (c)\;\log \bigg(\frac{1}{4}\bigg) \\ (d)\;None \end {array}\]

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1 Answer

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$\sin x=t, \qquad\:\:when\: x=0,\:\: t=0$
$\cos x dx=dt, \qquad\:when\:\: x=\large\frac{\pi}{2},$$\:\:t=sin\large\frac{\pi}{2}=$$1$
=>$\int \limits_0^1 \large\frac{dt}{(t+1)(t+2)}=> \int \limits_0^1 \frac{1}{t+1}$$dt -\int \limits_0^1 \large\frac{1}{t+2}$$dt$
=>$\bigg[\log (t+1)\bigg]_0^1-\bigg[ \log (t+2)\bigg]_0^1$
=>$\log 2- \log 0- \log 3+ \log 2$
=>$\log 4- \log 3$
=> $ \log \bigg(\large\frac{4}{3}\bigg)$
Hence a is the correct answer.


answered Dec 17, 2013 by meena.p
edited Dec 17, 2013 by rvidyagovindarajan_1

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