Evaluating integrals Evaluate the following integrals.

∫ 𝓍² cos 𝓍³ d𝓍

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Step 1: Recognize that the integral ∫𝓍² cos(𝓍³) d𝓍 involves a composite function. This suggests that substitution might be a useful method to simplify the integral.

Step 2: Let u = 𝓍³. Then, compute the derivative of u with respect to 𝓍: du/d𝓍 = 3𝓍², which implies du = 3𝓍² d𝓍.

Step 3: Rewrite the integral in terms of u. Substitute u = 𝓍³ and du = 3𝓍² d𝓍 into the integral. The integral becomes (1/3) ∫cos(u) du, where the factor of 1/3 comes from dividing by 3 to account for the substitution.

Step 4: Evaluate the integral of cos(u) with respect to u. The integral of cos(u) is sin(u). Thus, the integral becomes (1/3) sin(u) + C, where C is the constant of integration.

Step 5: Substitute back u = 𝓍³ to express the solution in terms of the original variable 𝓍. The final result is (1/3) sin(𝓍³) + 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 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.

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. For example, in the integral ∫ x² cos(x³) dx, one might let u = x³, which simplifies the integral significantly.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions often appear in integrals and derivatives. Understanding their properties, such as periodicity and symmetry, is crucial for evaluating integrals that involve trigonometric expressions, like cos(x³) in the given integral.

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