Evaluate the integrals in Exercises 31–78.
75. ∫(from -2...

Question

Evaluate the integrals in Exercises 31–78.

75. ∫(from -2 to -1)2dv/(v²+4v+5)

Accepted Answer

Welcome back everyone. Evaluate the definite integral, integral from 0 to 2 of 3 d x divided by x 2 minus 4 x plus 8. A pi divided by 4, B 3 pi divided by 8, C 5 pi divided by 4, and D 3 pi divided by 2. For this problem, let's consider the denominator first. It is x2 minus 4 x + 8. What we can do is complete the square and write it as x2 minus 4 x. + 4 + 4. And this allows us to complete the squares x minus 22 plus 4 so that the integral can be rewritten as the integral from 0 to 2 of we can factor out the constant which is 3. DX divided by x minus 22 + 4. And here we can use the use substitution u equals x minus 2. The derivative of u with respect to x is going to be the derivative of x minus 2, which is 1, which essentially means that dU is equal to dx. Now let's proceed with the new limits of integration. The lower one is going to be 0 minus 2, which is -2, and the upper limit of integration is going to be 2 minus 2, which is 0. So we can now write the integrals 3 integral from -2 up to 0 of dx becomes duu. And in the denominator we have U2 + 4, which is U2 + 22. And the reason why this is useful is because we know that the integral. Of d u divided by u2 plus a 2 is equal to 1 divided by a. Inverse of tangent of a u divided by a plus e constant of integration c. So let's apply it to our integral. Let's label it. With I It is going to be equal to 3 multiplied by 1 divided by a, which is 2 in this case. Inverse of tangent of u divided by a again a is 2 and we're evaluating the integral from -2 up to 0. Now all that we have to do is apply the evaluation theorem, which gives us 3 halves. In parent inverse of tangent of 0 divided by 2, which is 0. Minus inverse of tangent of 2 divided by 2 which is -1. And this is going to be equal to 3 halves. Inverse of tangent of 0 is 0 minus inverse of tangent of -1 is negative pi divided by 4. So we get 3 halves multiplied by pi divided by 4, which is equal to 3 pi divided by 8. And this corresponds to the answer of choice B. Thank you for watching.

Evaluate the integrals in Exercises 31–78.75. ∫(from -2 to -1)2dv/(v²+4v+5)

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Evaluate the integrals in Exercises 31–78.
75. ∫(from -2 to -1)2dv/(v²+4v+5)