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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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if $h(a)=h(b) $ then $ \int \limits_a^b [f(g(h(x)))]^{-1} . f ' (g (h(x)))g ' (h(x)).h'(x) dx=?$

\[\begin {array} {1 1} (a)\;0 \\ (b)\;f(a)-f(b) \\ (c)\;f[g(a)] \\ (d)\;None \end {array}\]

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Let $f[g(h(x)]=t$
$\Rightarrow\:f'[g(h(x))]. g'(h(x)].h'(x).dx$$=dt$
when $x=a$, $t=f[g(h(a))]\:\:and\:\:when \:\:x=b,\:\:t=f[g(h(b))]$
But given that $h(a)=h(b)\:\therefore\:when\:x=b,\:\:\:t=f[g(h(a))]$
Substituting the values in the given integral I we get
$I=\int \limits_ {f(g(h(a)]} ^ {f[g(h(a))]} \large\frac{1}{t}$$. dt$
$\Rightarrow\: \bigg[ \log t \bigg]_{f[g(h(a))]}^{f[(g)h(a)]}$


answered Dec 16, 2013 by meena.p
edited Dec 17, 2013 by rvidyagovindarajan_1

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