Find the definite integral of x*sqrt(x-1) on [1,2]: u-substitution and transforming the limits.

We use a u-substitution and transform the limits of integration to find the definite integral of x*sqrt(x-1) on the interval [1,2].

The standard thing we try first for an integral of this type is to let u equal the interior of the square root. So, we let u=x-1 and du=dx, but we have to take care of the x in the integrand. This is where we solve for x in our original substitution to get x=u+1, and then we can transform the entire integral in terms of u.

Next, we transform the limits of integration to u-space. So we look at the lower bound for x, that's x=1, and we plug into our substitution to find that u=0. Similarly, we plug in the upper limit of x=2 and find that u=1.

Now the entire integral is transformed to u-space and we compute the integral using the power rule and evaluating across the limits of integration.

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