7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a composite function, eˣ, inside another function sec(eˣ + 1). This suggests that substitution might be a useful technique.
2. Step 2: Let u = eˣ + 1. Then, compute the derivative of u with respect to x: du/dx = eˣ. This implies that du = eˣ dx.
3. Step 3: Rewrite the integral in terms of u. Substituting u = eˣ + 1 and du = eˣ dx, the integral becomes ∫ sec(u) du.
4. Step 4: Recall the standard integral formula for sec(u): ∫ sec(u) du = ln|sec(u) + tan(u)| + C, where C is the constant of integration.
5. Step 5: Substitute back u = eˣ + 1 into the result to express the solution in terms of x. The final expression will be ln|sec(eˣ + 1) + tan(eˣ + 1)| + C.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, recognizing the structure of the integrand can help determine the appropriate method to simplify the integral.

Substitution Method

The substitution method is a powerful technique in integration where a new variable is introduced to simplify the integral. By letting u = eˣ + 1, the integral can be transformed into a more manageable form. This method is particularly useful when the integrand contains a function and its derivative.

Exponential Functions

Exponential functions, such as eˣ, are functions of the form f(x) = a^x, where 'a' is a constant. They have unique properties, including the fact that their derivative is equal to the function itself. Understanding the behavior of exponential functions is crucial for evaluating integrals involving them, as they often appear in various calculus problems.