Evaluating integrals Evaluate the following integrals.

∫ sin 𝒵 sin (cos 𝒵) d𝒵

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Step 1: Recognize that the integral involves a composition of functions, specifically sin(𝒵) and sin(cos(𝒵)). This suggests that substitution might be a useful technique.

Step 2: Let u = cos(𝒵). Then, the derivative of u with respect to 𝒵 is du/d𝒵 = -sin(𝒵), or equivalently, du = -sin(𝒵) d𝒵.

Step 3: Rewrite the integral in terms of u. Substituting u = cos(𝒵) and du = -sin(𝒵) d𝒵, the integral becomes -∫ sin(u) du.

Step 4: Evaluate the integral of -sin(u) with respect to u. The antiderivative of sin(u) is -cos(u), so the integral becomes -(-cos(u)) = cos(u).

Step 5: Substitute back u = cos(𝒵) to return to the original variable. The final expression for the integral is cos(cos(𝒵)) + C, where C is the constant of integration.

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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 on a given interval. It can be thought of as the reverse process of differentiation. There are various techniques for integration, including substitution, integration by parts, and numerical methods, each suited for different types of functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to the ratios of sides in right triangles. In calculus, these functions are often encountered in integrals and derivatives. Understanding their properties, such as periodicity and symmetry, is crucial for evaluating integrals that involve these functions, as they can simplify the integration process.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. It is particularly useful when dealing with composite functions, such as those involving trigonometric functions, as it can transform the integral into a more manageable form.

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