Integrate the following(6x^5-2x+1)dx
just use the power rule and integrate each term separately.
recall that
∫x^n dx = 1/(n+1) x^(n+1)
you kidding?
x^6 - x^2 + x + c
To integrate the given expression, you need to apply the power rule of integration. This rule states that when integrating a term with the variable x raised to a power n, the result is (1/(n+1)) times the variable x raised to the power (n+1).
In this case, you have three terms: 6x^5, -2x, and 1. Let's integrate each term separately and add them up at the end.
1. Integrate 6x^5: To integrate 6x^5, apply the power rule. The power of x is 5, so we add 1 to the power and divide the coefficient by the new power.
∫ 6x^5 dx = (6/6)x^(5+1) + C = x^6 + C, where C is the constant of integration.
2. Integrate -2x: Using the power rule, integrate -2x. The power of x is 1.
∫ -2x dx = (-2/2)x^(1+1) + C = -x^2 + C.
3. Integrate 1: The integral of a constant is equal to the variable multiplied by the constant of integration.
∫ 1 dx = x + C.
Now, sum up all the integrals:
∫ (6x^5 - 2x + 1) dx = x^6 - x^2 + x + C, where C is the constant of integration.
Therefore, the integral of (6x^5 - 2x + 1) dx is x^6 - x^2 + x + C, where C is the constant of integration.