(e^x/x^2)

To integrate the function \( \frac{e^x}{x^2} \), we can use integration by parts. The formula for integration by parts is:

\[ \int u \, dv = uv - \int v \, du \]

Let's choose:

\( u = \frac{1}{x^2} \) => \( du = -\frac{2}{x^3} \)

\( dv = e^x \) => \( v = e^x \)

Now using the integration by parts formula:

\[ \int \frac{e^x}{x^2} \, dx = -\frac{e^x}{x^2} - \int -\frac{2e^x}{x^3} \, dx \]

Simplifying:

\[ \int \frac{e^x}{x^2} \, dx = -\frac{e^x}{x^2} + 2\int \frac{e^x}{x^3} \, dx \]

Now, we need to integrate \( \frac{e^x}{x^3} \). This can be done using integration by parts again:

Let's set:

\( u = \frac{1}{x^3} \) => \( du = -\frac{3}{x^4} \)

\( dv = e^x \) => \( v = e^x \)

Applying integration by parts here:

\[ \int \frac{e^x}{x^3} \, dx = -\frac{e^x}{x^3} + 3\int \frac{e^x}{x^4} \, dx \]

Continuing to solve the new integral and then integrating back up the chain, we can find the final result.