91Ó°ÊÓ

The substitution method, also known as u-substitution, is an essential technique for simplifying complex integrals. In this method, we make a substitution that transforms the integral into a more manageable form. For the given exercise, we substitute the expression \( u = \frac{3}{x} \), which simplifies the integration process.

This substitution is chosen because the derivative \( du = -\frac{3}{x^2} \, dx \) acts as a simpler replacement for part of the original integrand. This substitution transforms the integral in terms of \( x \) into an integral in terms of \( u \), making it easier to evaluate.

• Choose a substitution that will simplify the integrand.
• Make sure to replace both the function and the differentials.
• Rewrite the original integral, when possible, entirely in terms of \( u \).

Once the integral is rewritten, it often becomes straightforward to solve. The substitution method is a powerful tool in calculus for tackling integrals that seem difficult at first glance.

Finding the antiderivative is a crucial step in solving integrals. The antiderivative of a function is the reverse process of differentiation. For the transformed integral \( \int e^u \, du \), the antiderivative is directly \( e^u \).

When finding an antiderivative, remember that you're looking for a function whose derivative returns the original integrand. Here are a few points to consider:

• The antiderivative of \( e^u \) is \( e^u \), the same as the function itself.
• There is no change other than the variable of integration.
• Once the antiderivative is found, apply the integral bounds to evaluate the definite integral.

This exercise demonstrates the simplicity of finding an antiderivative when the integrand matches common functions such as the exponential function.

Integral bounds define the start and end points over which we evaluate a definite integral. In our problem, the original bounds were from \( x = 1 \) to \( x = 3 \). However, with substitutions, these bounds need adjustment.

When substituting, remember to convert the bounds from the original variable to the new variable. In this case:

• The substitution leads to bounds that correspond to \( x = 1 \) being \( u = 3 \) and \( x = 3 \) becoming \( u = 1 \).
• Switching integration bounds occurs due to the negative sign in \( du \), altering them to from \( u = 1 \) to \( u = 3 \).

Once substitution transforms the bounds, you can precisely evaluate over the corrected interval. Always double-check your substitution's impact on bounds to ensure accuracy.

Graphing utility verification serves as a practical means to validate calculated integrals. After analytically solving the problem, use a tool like a graphing calculator or software to graph the function \( \frac{e^{3/x}}{x^2} \) over the interval from \( x = 1 \) to \( x = 3 \).

This visual approach verifies the definite integral's result by showing the area under the function's curve matches the compute value of \( e - e^3 \).

• Input the function into the graphing tool.
• Set the correct interval for \( x \).
• Observe the graph to ensure the area matches the previously found definite integral value.

Graphing verification combines visual and analytical skills, strengthening understanding and confirming the manual calculations are correct.