Understanding Integration of cf(x) with Respect to x and the Power Rule

∫ cf(x)dx

To find the integral of cf(x) with respect to x, where c is a constant, we can use the power rule of integration

To find the integral of cf(x) with respect to x, where c is a constant, we can use the power rule of integration. The power rule states that if we have a function of the form x^n, where n is any real number (except -1), the integral of x^n with respect to x is (x^(n+1))/(n+1), where (n+1) is the power of the x term.

In the given function, cf(x), c is a constant and f(x) is a general function. Since c is a constant, it can be pulled outside of the integral. Hence, we have:

∫ cf(x)dx = c ∫ f(x)dx

Now, you can integrate f(x) using the appropriate integration technique, depending on the form of the function f(x).

For example, if f(x) is a polynomial function, you can apply the power rule of integration for each term in the polynomial. If f(x) is a trigonometric function, you might have to use specific trigonometric integration formulas.

In any case, once you integrate the function f(x), you can then multiply it by the constant c to obtain the final result.

Remember to always include the constant of integration (usually represented by “+ C”) when finding indefinite integrals, as it accounts for the possibility of an overall shift in the function.

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