Evaluate the integrals in Exercises 25–30 by using a substitut...

Question

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ from 0 to 1 x√(1 - x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral from 0 to 1 x square root of 1 minus x2 dx. For this problem we're going to use the U substitution. Let's suppose that U is equal to 1 minus X2. Let's remember that our next step is to find the derivative of u with respect to x, and that's the derivative of 1 minus x 2, which is -2 x. Since we have x d x in the integrant, we're going to solve for x multiplied by dx. And in order to get that expression, let's multiply both sides by dx and divide both sides by -2, which gives us x multiplied by dx equals. 1/2 the EU. Now let's change the limits of integration, so the lower 1 u1 is 1 minus 02, which is 1, and the upper limit u2 is going to be 1 minus 12, which is 0. So now we can rewrite the integral as integral from 1 up to 0 of square root of u followed by xdx, which is 1/2 duu. We can factor out -1/2 and simply ride the. Now let's apply the reverse interval property. We're going to turn -1/2 into 1/2 and swap the limits of integration from 0 to 1. And now in order to integrate square root of you, we're going to write it as to use the power of 1/2, right? This allows us to apply the power rule, and we're going to get 1/2 multiplied by u to the power of 1/2 plus 1, which is 3 halves, divided by 3 halves from 0 to 1. Now let's apply the fundamental theorem of calculus part 2, which gives us 1/2 multiplied by 2/3. We can. Simplify right and evaluate you to the power of 3 halves. From 0 to 1. This is equal to 1/3. And in parentheses we take 1 312 minus 0 31. Which is equal to 1/3 multiplied by 1 minus 0, that's 1, so we basically get 1/3 as the final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.∫ from 0 to 1 x√(1 - x) dx

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Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx