Use the table of integrals at the back of the text to evaluate...

Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x (7x + 5)^(3/2) dx

Accepted Answer

Welcome back, everyone. Using a standard integral table, evaluated the following indefinite integral. Integral of X multiplied by 4 X plus 3 to the power of 3 halves DX. For this problem, we're going to use the formula integral of X multiplied by AX plus B. Race to the power of MDX is equal to AX plus B. Raise the power of N + 1. Divided by a squared. Multiplied by A X plus B divided by M + 2. And minus B divided by N + 1 plus a constant of integration C, where N is not equal to -1 and not equal to -2. So, in this problem, by analogy, we can see that A is equal to 4, B is the free coefficient, which is 3. And N is our exponent. It is equal to 3 halves. So we can now show that the integral of X multiplied by 4 X plus 3. So the power of 3 halves, the X is going to be equal to. First of all, we're going to take A X + B, so that's 4 X + 3 to the power of N + 132 + 1 gives us an exponent of 5 halves. We are now dividing by a squared, which is 4 squad or basically 16. And this is multiplied by A X + B, so 4 X + 3, divided by N + 2. So, 3 halves plus 2, which gives us seven halves, right? Minus B, which is 3 divided by n + 1. And plus 1 is 5 halves. We already know that. And we're adding a constant of integration C. So now, our goal is to simplify this expression. Let's go ahead and do that. So we're going to get 4 X plus 3 raise to the power of 5 halves. Divided by 16, multiplied by dividing by 7 halves, we get 2 multiplied by 4 X + 3 divided by 7, and we're subtracting 3 multiplied by 2 divided by. 5, which is 6 divided by 5 plus a constant of integration C. So now, let's find the common denominator for the factor in parent, right? We have 7 multiplied by 5, which is 35, so we're going to rewrite 4 X + 3. So the power of 5 halves divided by 16. Multiplied by using the common denominator, we can write 35, right? And now 2 is going to be multiplied by 5 to give us. 10 multiplied by 4 X + 3 minus 6 multiplied by 7, right? So now distributing, we're going to get 40 X plus 30 minus 42. We can write. 40 X. Plus 30 minus 42, so we can just say minus 12, combining the like terms. And now, what we're going to do from here is perform a factorization to simplify the expression, right? Let's go ahead and factor out 4. So that we get 4 in 10 X minus 3, and then we can cancel out that 4 because we have 16 within the previous factor, right? 16 divided by 4 gives us 4 in the denominator. And then we can go ahead and write our final answer, that would be. 10 x minus 3 multiplied by 4 X plus 3. So the power of five halves. We're also going to add a constant of integration, right? Divided by 4 multiplied by 35 gives us 140, and we're going to add the constant of integration for an indefinite integral, and that's our final answer for this problem. Thank you for watching.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ x (7x + 5)^(3/2) dx

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx