Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫₀³ (x + 2)√(x + 1) dx

Accepted Answer

Welcome back everyone to another video. Evaluate the integral from 0 to 3 of x + 3 multiplied by the square root of x + 1 dx. Option A says 352 divided by 5, B 326 divided by 5, C 326 divided by 15, and D 352 divided by 15. We're going to solve this problem using U substitution. Let's suppose that u is equal to x + 1. Additionally, we can express x + 3 if we add 2 to both sides of this equation, so that u + 2 is going to be x + 3. Let's remember that our next step is to find the derivative of u with respect to x, which is the derivative of x plus 1, and that's 1, which essentially shows us that duu is equal to dx because their ratio is equal to 1. And now let's find the new limits of integration. So the lower 1 u1 is 0 + 1, which is 1, and the upper limit of integration u2 is 3 + 1, which is 4. So now we can rewrite the integral as the integral from 1 up to 4. X + 3 becomes U + 2. And the square root of x + 1 becomes square root of u. Dx becomes duu. Well then, what we can do now is rewrite the square root of you as you raise the power of 1/2 and distribute. So we're going to have integral from 1 up to 4 of u to the power of 3 halves plus 2 u to the power of 1/2 the u. And from here we can integrate using the power rule which gives us u to the power of 3 halves plus 1, that's u to the power of 5 halves divided by 5 halves or basically multiplied by 2/5, right, plus 2, you raise to the power of 1/2 plus 1, which is 3 halves, and we're dividing by 3 halves or simply multiplying by 2/3, right? And we want to evaluate the result from 1 up to 4. Let's go ahead and simplify the function slightly. What we're going to do is simply rewrite it as 2/5. From here we can rewrite you to the power of 5 halves as u squared square root of u. And for the next term, we have 4. Thirds U to the power of 3 halves, which can be written as u square root of u. Let's add the limits of integration, that's from 1 to 4. And additionally, we can perform. At least partial factorization, right? So what we're going to do is simply factor out u, square root of u for simplicity. And in parentheses we will have 2/5. You Plus 43. We want to evaluate from 1 to 4, and this is where we start applying the evaluation theorem. So first of all, we will have 4 multiplied by the square root of 4. Multiplied by 2/5 multiplied by 4 + 4/3. And we're going to subtract. 1 multiplied by the square root of 1 multiplied by 2/5 multiplied by 1 + 4/3. Now, let's perform algebraic steps and simplify. So that's 4 multiplied by 2, which is 8. In parentheses we have. 8th 5th. Plus 4/3. Let's find the common denominator that'll be 15. So 8/5 would be 24 divided by 15, and for 4/3, we are going to have 20 divided by. 15, right? 24 plus 20 is 44, so we basically end up with 44 divided by 15. And we're going to subtract. One multiplied by square root of one that's basically one. So we're going to subtract in parentheses. 2/5 Plus 4/3, right? So again, we're looking at the common denominator, which is 15, so the first fraction becomes. 6 divided by 15 and for the second one, that's again 20 divided by 15. So we get 26 divided by 15, right, meaning we're subtracting. 26 divided by 15 for now, let's not. Simplify yet let's simply factor out. So we're going to have 1 divided by 15, let's simply factor out. The denominator and in parentheses we will have 8 multiplied by 44 minus 26. So we will have 15 in the denominator in the numerator that's 352 minus 26. Subtracting, we're going to get 326. Divided by 15, which corresponds to the answer of choice c. Thank you for watching.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫₀³ (x + 2)√(x + 1) dx

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₀³ (x + 2)√(x + 1) dx