Evaluating integrals Evaluate the following integrals.

∫π/₁₂^π/⁹ (csc 3𝓍 cot 3𝓍 + sec 3𝓍 tan 3𝓍) d𝓍

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Step 1: Recognize the integral components. The given integral is ∫π/₁₂^π/⁹ (csc(3𝓍)cot(3𝓍) + sec(3𝓍)tan(3𝓍)) d𝓍. Notice that the terms csc(3𝓍)cot(3𝓍) and sec(3𝓍)tan(3𝓍) are derivatives of specific trigonometric functions.

Step 2: Simplify each term using trigonometric identities. Recall that the derivative of -csc(3𝓍) is csc(3𝓍)cot(3𝓍), and the derivative of sec(3𝓍) is sec(3𝓍)tan(3𝓍). This allows us to rewrite the integral as ∫π/₁₂^π/⁹ (-d(csc(3𝓍)) + d(sec(3𝓍))).

Step 3: Apply the property of definite integrals. Since the integral is now expressed in terms of derivatives, it simplifies to evaluating the antiderivatives at the bounds. Specifically, evaluate -csc(3𝓍) and sec(3𝓍) at the limits π/₁₂ and π/⁹.

Step 4: Substitute the limits into the antiderivatives. Compute -csc(3𝓍) and sec(3𝓍) at 𝓍 = π/₁₂ and 𝓍 = π/⁹. Use the fundamental theorem of calculus to find the difference between the values at the upper and lower limits.

Step 5: Combine the results. Add the evaluated values of -csc(3𝓍) and sec(3𝓍) at the respective limits to obtain the final result of the integral.

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Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate definite integrals, which provide a numerical value over a specified interval, or indefinite integrals, which yield a family of functions. Understanding the properties and techniques of integration is essential for evaluating integrals effectively.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, secant, cosecant, tangent, and cotangent, are crucial in calculus, especially when dealing with integrals involving angles. These functions have specific identities and relationships that can simplify the integration process. Recognizing how to manipulate these functions and apply their identities is key to solving integrals that involve trigonometric expressions.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This method involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. Mastery of substitution is important for tackling complex integrals, especially those involving composite functions or trigonometric identities.

Evaluating integrals Evaluate the following integrals.∫π/₁₂^π/⁹ (csc 3𝓍 cot 3𝓍 + sec 3𝓍 tan 3𝓍) d𝓍