Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (1 + tan x) sec²x dx

Verified step by step guidance

1. Step 1: Observe the integrand ∫ (1 + tan x) sec²x dx. Notice that it contains a combination of trigonometric functions: tan(x) and sec²(x). Recall that the derivative of tan(x) is sec²(x), which suggests a potential substitution method.
2. Step 2: Let u = tan(x). Then, the derivative of u with respect to x is du/dx = sec²(x), or equivalently, du = sec²(x) dx. This substitution simplifies the integral by replacing sec²(x) dx with du.
3. Step 3: Rewrite the integrand in terms of u using the substitution u = tan(x). The integral becomes ∫ (1 + u) du.
4. Step 4: Simplify the new integral ∫ (1 + u) du into two separate terms: ∫ 1 du + ∫ u du. This makes the integration process straightforward.
5. Step 5: Identify the integration technique as basic integration of polynomial terms. The integral ∫ 1 du evaluates to u, and ∫ u du evaluates to u²/2. Combine these results after integration to express the solution in terms of u, and then back-substitute u = tan(x) to return to the original variable.

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. Understanding these methods allows one to choose the most effective approach based on the form of the integrand.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. For example, the identity sec²x = 1 + tan²x can simplify integrals involving secant and tangent functions. Recognizing and applying these identities can make integration more straightforward.

Simplifying the Integrand

Simplifying the integrand involves rewriting it in a form that is easier to integrate. This may include factoring, combining like terms, or using trigonometric identities. A simpler integrand can often lead to a more direct application of integration techniques, making the evaluation process more efficient.