find the integral of (a^x+b^x)^2/a^xb^x. solve the same

To solve the integral of the function \(\frac{(a^x + b^x)^2}{a^x b^x}\), we can start by simplifying the expression. The goal is to rewrite the integrand in a way that makes it easier to integrate. Let's break it down step by step.

Simplifying the Integrand

First, we can rewrite the integral as follows:

\[\int \frac{(a^x + b^x)^2}{a^x b^x} \, dx = \int \left( \frac{(a^x + b^x)^2}{a^x} \cdot \frac{1}{b^x} \right) \, dx\]

Expanding the Square

Next, we expand \((a^x + b^x)^2\):

\[(a^x + b^x)^2 = a^{2x} + 2a^x b^x + b^{2x}\]

Substituting this back into our integral gives:

\[\int \frac{a^{2x} + 2a^x b^x + b^{2x}}{a^x b^x} \, dx\]

Breaking Down the Integral

Now, we can split the integral into three separate parts:

\[\int \left( \frac{a^{2x}}{a^x b^x} + 2 \frac{a^x b^x}{a^x b^x} + \frac{b^{2x}}{a^x b^x} \right) \, dx = \int \left( \frac{a^{2x}}{b^x} + 2 + \frac{b^{2x}}{a^x} \right) \, dx\]

Further Simplification

Now, we can rewrite the integral as:

\[\int \left( a^{x} b^{-x} + 2 + b^{x} a^{-x} \right) \, dx\]

Integrating Each Term

At this stage, we can integrate each term individually:

• For the term \(a^{x} b^{-x}\):

We can rewrite it as \(\left(\frac{a}{b}\right)^{x}\). The integral becomes:

\[\int \left(\frac{a}{b}\right)^{x} \, dx = \frac{b}{\ln\left(\frac{a}{b}\right)} \left(\frac{a}{b}\right)^{x} + C_1\]

• For the constant term 2:

The integral is straightforward:

\[\int 2 \, dx = 2x + C_2\]

• For the term \(b^{x} a^{-x}\):

This can be rewritten as \(\left(\frac{b}{a}\right)^{x}\), and the integral becomes:

\[\int \left(\frac{b}{a}\right)^{x} \, dx = \frac{a}{\ln\left(\frac{b}{a}\right)} \left(\frac{b}{a}\right)^{x} + C_3\]

Combining Results

Now, let's combine all the results from our integrations:

\[\int \frac{(a^x + b^x)^2}{a^x b^x} \, dx = \frac{b}{\ln\left(\frac{a}{b}\right)} \left(\frac{a}{b}\right)^{x} + 2x + \frac{a}{\ln\left(\frac{b}{a}\right)} \left(\frac{b}{a}\right)^{x} + C\]

Here, \(C\) is the constant of integration that combines \(C_1\), \(C_2\), and \(C_3\). This gives us the final expression for the integral. If you have any further questions or need clarification on any of the steps, feel free to ask!