Evaluate the integrals in Exercises 31–78.
37. ∫(from -1...

Question

Evaluate the integrals in Exercises 31–78.

37. ∫(from -1 to 1)dx/(3x-4)

Accepted Answer

Welcome back everyone. Evaluate the integral dx divided by 2 x plus 1 from 0 to 2. A 2 LN5, B 1/5 LN2, C 5 LN2, and D 1/2 LN5. We're going to approach this problem using U substitution. Let's suppose that U is equal to 2 x plus 1, and let's remember that the next step is to find the derivative of u with respect to x. There's the derivative of 2 x plus 1, which is 2. So now what we can do is simply show that DX is equal to DU divided by 2 or 1/2 DU. Now, the next step is to change the limits of integration, so the lower one is going to be 2 multiplied by 0 + 1. And that's one And the upper one is going to be 2 multiplied by 2 + 1, which is 5. Now, we are ready to rewrite the integral in terms of UNDU. We're integrating from 1 up to 5. DX becomes 1/2 DU, so we can write DU and we can factor out 1/2. And in the denominator we simply have you. Notice that this is an elementary integral, so we get our constant 1 half. The integral of d u divided by u is ln of the absolute value of u, but we can just write ln of u because u is between 1 and 5, and these are positive numbers. And we want to evaluate the result from 1 up to 5. Applying the evaluation theorem we can factor out 1/2 and in parentheses we're going to evaluate ln at 5 and subtract ln at 1. Since LN of 1 is here, we can just cross out that term. And show that it is equal to 0 and this basically gives us the final answer, right 1/2. LM 5 which corresponds to the answer of choice d. Let's label it and thank you for watching.

Evaluate the integrals in Exercises 31–78.37. ∫(from -1 to 1)dx/(3x-4)

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Evaluate the integrals in Exercises 31–78.
37. ∫(from -1 to 1)dx/(3x-4)