Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx

Guida verificata passo dopo passo

Identify the integral to solve: \(\int \frac{e^{1/x}}{x^{2}} \, dx\).

Look for a substitution that simplifies the exponent and the denominator. Notice that the exponent is \(\frac{1}{x}\), so let \(u = \frac{1}{x}\).

Compute the differential \(du\) in terms of \(dx\): Since \(u = x^{-1}\), then \(du = -x^{-2} \, dx\), or equivalently, \(-du = x^{-2} \, dx\).

Rewrite the integral in terms of \(u\) and \(du\): Substitute \(e^{1/x} = e^{u}\) and \(x^{-2} \, dx = -du\), so the integral becomes \(\int e^{u} (-du) = -\int e^{u} \, du\).

Integrate with respect to \(u\): The integral of \(e^{u}\) is \(e^{u}\), so the integral becomes \(-e^{u} + C\). Finally, substitute back \(u = \frac{1}{x}\) to express the answer in terms of \(x\).

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