69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.

Verified step by step guidance

1. Step 1: Identify the substitution given in the problem. Here, we are instructed to use u = sec(x). This substitution will help simplify the integral.
2. Step 2: Compute the derivative of u with respect to x. Since u = sec(x), we have du/dx = sec(x)tan(x). Therefore, du = sec(x)tan(x) dx.
3. Step 3: Rewrite the integral ∫(tan(x) sec²(x)) dx using the substitution u = sec(x). Replace sec(x) with u and tan(x) sec(x) dx with du.
4. Step 4: After substitution, the integral becomes ∫u du because sec²(x) becomes u² and tan(x) sec(x) dx is replaced by du.
5. Step 5: Integrate ∫u du to find the antiderivative. The result will be expressed in terms of u, and then substitute back u = sec(x) to express the final answer in terms of x.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for solving problems involving accumulation and area, and it often requires techniques such as substitution to simplify the integrand.

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. In this case, using the substitution u = sec(x) allows us to express the integral in terms of u, making it easier to evaluate.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving integrals involving trigonometric functions. In this problem, recognizing the relationship between tan(x), sec(x), and their derivatives is key to applying the substitution effectively.