Evaluating the Integral ∫ tan^2 x sec^3 x dx

Evaluating integrals is a fundamental skill in calculus, and the integral ∫ tan^2 x sec^3 x dx is a classic example that showcases the application of various integration techniques. In this article, we will walk through the step-by-step process of evaluating this integral, highlighting the key concepts and strategies involved.

Step 1: Understand the Integrand

The integrand in this case is the product of two trigonometric functions: tan^2 x and sec^3 x. To evaluate this integral, we need to recognize the patterns and relationships between these functions.

Step 2: Apply the Power Rule

The power rule for integration states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. In our case, we can apply this rule to the sec^3 x term, as sec x is the derivative of tan x.

Step 3: Utilize the Substitution Method

To simplify the integration, we can apply the substitution method. Let u = tan x, then du = sec^2 x dx. Substituting these values, we get:

∫ tan^2 x sec^3 x dx = ∫ (u^2) (sec^2 x) (sec x) du

Step 4: Evaluate the Integral

Now that we have the substituted expression, we can evaluate the integral using the power rule:

∫ (u^2) (sec^2 x) (sec x) du = ∫ u^3 du = (u^4)/4 + C

Step 5: Substitute Back

Finally, we substitute the original variable back into the result:

∫ tan^2 x sec^3 x dx = (tan^4 x)/4 + C

Therefore, the solution to the integral ∫ tan^2 x sec^3 x dx is (tan^4 x)/4 + C, where C is the constant of integration.

This evaluation process demonstrates the application of fundamental integration techniques, such as the power rule and the substitution method, to solve a complex integral involving trigonometric functions. Understanding these techniques is crucial for mastering calculus and solving a wide range of mathematical problems.