Evaluating integrals Evaluate the following integrals.


∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) d𝓍

Verified step by step guidance

1. Step 1: Break the integral into separate terms. Using the linearity property of integrals, rewrite the given integral as: ∫₋π/₂^π/₂ cos(2𝓍) d𝓍 + ∫₋π/₂^π/₂ cos(𝓍)sin(𝓍) d𝓍 ― ∫₋π/₂^π/₂ 3sin(𝓍⁵) d𝓍.
2. Step 2: Evaluate the first term ∫₋π/₂^π/₂ cos(2𝓍) d𝓍. Use the substitution method where u = 2𝓍, and du = 2 d𝓍. Adjust the limits of integration accordingly.
3. Step 3: For the second term ∫₋π/₂^π/₂ cos(𝓍)sin(𝓍) d𝓍, recognize that cos(𝓍)sin(𝓍) can be rewritten as (1/2)sin(2𝓍) using the double-angle identity sin(2𝓍) = 2sin(𝓍)cos(𝓍). Substitute this into the integral.
4. Step 4: For the third term ∫₋π/₂^π/₂ 3sin(𝓍⁵) d𝓍, note that the integrand involves sin(𝓍⁵), which is not elementary. Consider symmetry properties of the sine function over the interval [-π/2, π/2] to simplify the evaluation.
5. Step 5: Combine the results of the three integrals to express the final solution. Ensure that any constants or simplifications are applied correctly.

Key Concepts

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 representing the area between the curve and the x-axis over a specified interval.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential in calculus for evaluating integrals involving periodic functions. Understanding their properties, such as their ranges and symmetries, is crucial for simplifying integrals and applying integration techniques effectively, especially when dealing with products of trigonometric functions.

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals by changing variables. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form, making it easier to evaluate. This method is particularly useful when dealing with composite functions or when integrating products of functions.