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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $\int \large\frac{(a^x -b^x)}{a^x b^x}$$dx$

\[\begin {array} {1 1} (a)\;\frac{(a/b)^x}{\log (a/b)}+\frac{(b/a)^x}{\log (b/a)}-2x+c \\ (b)\;\bigg(\frac{a/b}{\log (a/b)}\bigg)^x+\bigg(\frac{b/a}{\log (b/a)}\bigg)^x-2x+c \\ (c)\;\bigg(\frac{a}{b}\bigg)^x+\bigg(\frac{b}{a}\bigg)^x -2x+c \\ (d)\;None \end {array}\]

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A)
$\int \large\frac{a^{2x} }{(ab)^x} +\int \large\frac{b^{2x} }{(ab)^x}-\int \large\frac{2(ab)^{x} }{(ab)^x}$
$\int a^x b^{-x} dx+\int (\large\frac{b}{a} )^x $$dx- 2\int dx$
=> $\int \bigg(\large\frac{a}{b}\bigg)^x$$ dx+\int \bigg(\large\frac{b}{a}\bigg)^x$$dx-2 \int dx$
$ \large\frac{(a/b)^x}{\log (a/b)}+\frac{(b/a)^x}{\log (b/a)}$$-2x+c$
Hence a is the correct answer.
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