Problem 10 - Essential Calculus

Question

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \frac{e^{1 / t}}{t^{2}} d t $$

Step-by-Step Solution

Answer

The integral of \( \frac{e^{1/t}}{t^{2}} dt \) equals to \( -e^{-1/t}+C \).

Step 1: Setting up the integral

The integral to solve is given by: \[ \int \frac{e^{1 / t}}{t^{2}} d t \]

Step 2: Perform a substitution

To simplify the integral, we can perform a substitution. We choose -1/t = u, then du = dt/t^2. After the substitution, our integral transforms into:\[-\int e^u du\]

Step 3: Solve the integral

Now, the integral can be solved straightforwardly, since the integral of e^u is e^u. So we get:\[-e^u+C\]where C is the integration constant.

Step 4: Substitute back for t

In the last step, we substitute -1/t back for u to get the solution in terms of t:\[-e^{-1/t}+C\]That's the solution of the given integral.