Evaluating integrals Evaluate the following integrals.

∫(√1 + tan 2t) sec² 2t dt

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Step 1: Recognize that the integral involves a composite function. The term √(1 + tan(2t)) suggests a substitution might simplify the integral. Let u = 1 + tan(2t).

Step 2: Differentiate u with respect to t. Since u = 1 + tan(2t), we have du/dt = 2 sec²(2t). Therefore, du = 2 sec²(2t) dt.

Step 3: Rewrite the integral in terms of u. Substitute u = 1 + tan(2t) and du = 2 sec²(2t) dt into the integral. This gives (1/2) ∫√u du.

Step 4: Solve the simplified integral. The integral of √u is (2/3)u^(3/2). So, (1/2) ∫√u du becomes (1/3)u^(3/2) + C, where C is the constant of integration.

Step 5: Substitute back u = 1 + tan(2t) to return to the original variable. The final expression is (1/3)(1 + tan(2t))^(3/2) + C.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. There are various techniques for integration, including substitution, integration by parts, and recognizing standard forms. Understanding how to manipulate and evaluate integrals is crucial for solving problems in calculus.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus, especially when dealing with integrals involving angles. In this context, the integral provided includes the tangent function and its derivative, sec²(2t). Familiarity with the properties and identities of trigonometric functions is necessary for simplifying and evaluating integrals that involve these functions.

Substitution Method

The substitution method is a technique used in integration to simplify the process of finding an integral. It involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. In the given integral, recognizing that the derivative of the inner function (2t) is present can lead to a successful substitution, allowing for a more straightforward evaluation of the integral.

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