Question

The value of $\int \frac{(a^x+b^x{)}^2}{a^x\cdot b^x}dx$ is, where $a>1,b>1$ & $a\ne b$
​​​​​​​(where $C$ is constant of integration)

• A. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$
• B. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}-\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$
• C. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+x^2+C$
• D. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}-2x+C$

Answer 1

The correct option is A $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$

$\int \frac{(a^x+b^x{)}^2}{a^x\cdot b^x}dx=\int \frac{a^{2x}+b^{2x}+2(ab{)}^x}{a^x\cdot b^x}dx$
$=\int {\left(\frac{a}{b}\right)}^{x}dx+\int {\left(\frac{b}{a}\right)}^{x}dx+\int 2dx\{\because \int a^{x}dx=\frac{a^x}{\ln a}+C\}=\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$