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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Evaluate: $\int \frac{\left(a^x+b^x\right)^2}{a^x b^x} d x$
  • ✓
    $\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b$
  • B
    $\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{a}{b}}+2 x+C, a \neq b$
  • C
    $\left(\frac{a}{b}\right)^x+\left(\frac{b}{a}\right)^x+2 x+C, a \neq b$
  • D
    None of these
✓

Answer

Correct option: A.
$\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b$
(a) : We have, $\int \frac{\left(a^x+b^x\right)^2}{a^x b^x} d x=\int \frac{a^{2 x}+b^{2 x}+2 a^x b^x}{a^x b^x} d x$
$
=\int\left(\left(\frac{a}{b}\right)^x+\left(\frac{b}{a}\right)^x+2\right) d x=\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b
$

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