The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₁² (8 dx / (x² - 2x + 2))

Accepted Answer

Welcome back everyone. Compute the integral from -1 up to 0 of 4 d x divided by x2 + 2 x + 2. A says pi divided by 4. B says pi divided by 2, C, and D 2 pi. For this problem, we're going to consider the denominator first and we're going to complete the square. We can write x2 + 2 x + 1. It is a perfect square plus 1, right? Because we have + 2. And basically we can rewrite it as x + +12 + 1. So now what we can do is factor out 4 from the integral, and that'll be 4 integral from -1 up to 1 of the x. Divided by And now we are going to rewrite the denominator as x + 12 + 1. And from here we're going to use u substitution. Let's suppose that u is equal to x + 1. The derivative of U with respect to X is going to be the derivative of X plus 1, which is 1, meaning DU is equal to DX, right? So we can write that as well. And now we're going to convert the limits of integration. The lower limit of integration u1 is going to be -1 + 1, which is 0, and the upper limit of integration is going to be 0 + 1, which is 1. Now we are ready to rewrite the integral in terms of u and d u, so that would be 4. Integral from 0 to 1 of DU because dx is DU. Divided by U2 + 1 and we can notice that this is now an elementary integral. It is equal to inverse of tangent, so we basically get 4 inverse of tangent. Of you From 0 to 1. And all that we have to do is apply the fundamental theorem of calculus part 2, that be 4 in parenthes inverse of tangent of 1 minus inverse of tangent of 0. We get 4 in parentheses pi divided by 4, that's inverse of tangent of 1 minus 0, which is inverse of tangent of 0. That said, we get 4 pi divided by 4, which is pi. So the correct answer is option C. Thank you for watching.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫₁² (8 dx / (x² - 2x + 2))

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₁² (8 dx / (x² - 2x + 2))