Evaluating integrals Evaluate the following integrals.                                                                                                                                         
                                                                                                                                                                   
 ∫ (9𝓍⁸―7𝓍⁶) d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ (9𝓍⁸ ― 7𝓍⁶) d𝓍 is a polynomial integral, which can be solved term by term using the power rule for integration.
2. Step 2: Apply the power rule for integration to the first term, 9𝓍⁸. The power rule states that ∫ 𝓍ⁿ d𝓍 = (𝓍ⁿ⁺¹)/(n+1) + C, where n is the exponent. For 9𝓍⁸, the integral becomes (9𝓍⁹)/9.
3. Step 3: Apply the power rule for integration to the second term, -7𝓍⁶. Using the same rule, the integral becomes (-7𝓍⁷)/7.
4. Step 4: Combine the results from Step 2 and Step 3 into a single expression. Remember to include the constant of integration, C, at the end.
5. Step 5: Simplify the coefficients in the combined expression to finalize the integral in its simplified form.

Key Concepts

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 is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. The integral can be definite, with specific limits, or indefinite, representing a family of functions.

Power Rule for Integration

The Power Rule for Integration is a specific technique used to integrate polynomial functions. It states that the integral of x raised to the power n (where n is not equal to -1) is given by (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule simplifies the process of integrating polynomials by allowing for straightforward application to each term.

Constant Multiplication in Integration

When integrating a function that includes a constant multiplied by a variable term, the constant can be factored out of the integral. This means that if you have a function of the form k*f(x), where k is a constant, the integral can be expressed as k*∫f(x)dx. This property simplifies the integration process and allows for easier calculations.