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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $\int \large\frac{3}{7} .\frac{2^{\Large \log e^{x^2}}}{x}.$$dx$

$(a)\;\frac{3}{7} \log_2 2 \times 2 \log_e x^2 +c \\(b)\;(x^2-1)+c \\(c)\; \log_e^{x^2-1}+c \\ (d)\;\frac{3}{7} \frac{1}{\log_e 2} \times 2 ^{\log_e x^2} +c$
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1 Answer

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$\log _ex^2=t$
differentiate with respect to x
$\large\frac{1}{x^2} $$\times 2x dx =dt$
$\large\frac{2}{x} $$dx=dt$
=> $\int \large\frac{3}{7} . \frac{2t}{2} $$dt$
=> $ \large\frac{3}{7} \times \frac{1}{2} \bigg(\large\frac{2^t}{\log _e 2} \bigg)+c$
=> $ \large\frac{3}{7} \times \frac{1}{\log_e 2} $$ \times 2^{t-1} +c$
$\large\frac{3}{7} \frac{1}{\log_e 2}$$ \times 2 ^{\large\log_e x^2} +c$
Hence d is the correct answer.
answered Dec 31, 2013 by meena.p
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