find an antiderivative F(x) of f(x)=11x-sqrt(x)
∫(11x-sqrt(x))dx
=∫(11x-x^(1/2))dx
=(11/2)x²-(2/3)x^(3/2) + C
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To find an antiderivative, F(x), of f(x) = 11x - sqrt(x), we can integrate each term separately.
The integral of 11x with respect to x is (11/2) * x^2, where we add one to the power and divide by the new power.
Next, we want to integrate sqrt(x) with respect to x. We can rewrite sqrt(x) as x^(1/2), which gives us the integral as (2/3) * x^(3/2), applying the same logic explained above.
Finally, we can add these results together to find the antiderivative:
F(x) = (11/2) * x^2 + (2/3) * x^(3/2) + C
where C is the constant of integration.
To find the antiderivative of a function, you need to follow certain rules and techniques. In this case, we want to find the antiderivative of the function f(x) = 11x - sqrt(x).
Step 1: Start with the antiderivative of each term separately.
The antiderivative of a power of x is obtained by adding 1 to the exponent and dividing by the new exponent. The antiderivative of x^n is (x^(n+1))/(n+1).
The antiderivative of 11x is (11/2)x^2, obtained by adding 1 to the exponent of x (which is 1) and dividing by the new exponent (2).
The antiderivative of sqrt(x) is (2/3)x^(3/2), obtained by adding 1 to the exponent of x (which is 1/2) and dividing by the new exponent (3/2).
Step 2: Combine the antiderivatives of each term.
Adding the antiderivative of 11x and the antiderivative of sqrt(x), we get:
F(x) = (11/2)x^2 + (2/3)x^(3/2) + C
where C is the constant of integration, which accounts for the possibility that the original function had a constant term.
Therefore, the antiderivative of f(x) = 11x - sqrt(x) is F(x) = (11/2)x^2 + (2/3)x^(3/2) + C.