how do you solve:
integral:(x-1)sqrt(x) dx
There aren't any limits of integration
Add the integrals of
x^3/2 dx and -x(1/2) dx. Use the rule that the integral of x^n is
x^(n+1)/(n+1), where n is any constant.
To solve the integral ∫(x - 1)√(x) dx, you need to remember a few integration techniques. In this case, you'll need to use the power rule for integration, as well as the technique of performing integration by parts.
First, let's expand the integrand:
∫(x - 1)√(x) dx = ∫x√(x) dx - ∫√(x) dx
To solve the first integral, ∫x√(x) dx, you can use the power rule for integration. According to the power rule, the integral of x^n dx is (x^(n+1))/(n+1), where n is any constant.
In this case, n = 1/2. So, the integral of x√(x) dx becomes:
∫x√(x) dx = (x^(1/2+1))/(1/2+1) = (2x^(3/2))/(3/2) = (2/3)x^(3/2)
Next, let's solve the second integral, ∫√(x) dx. This integral is a little simpler, as it does not involve any variables besides x. The integral of √(x) dx can be directly computed as:
∫√(x) dx = (2/3)x^(3/2)
Note that the constant of integration, denoted by + C, is added at the end of each integral.
Therefore, the final solution to the integral is:
∫(x - 1)√(x) dx = ∫x√(x) dx - ∫√(x) dx = (2/3)x^(3/2) - (2/3)x^(3/2) + C = (2/3)x^(3/2) + C
In the absence of specific limits of integration, the answer will be in terms of x and the constant of integration, C.