Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 ...

Question

Evaluate the integrals in Exercises 31–78.

39. ∫(from 0 to π)tan(x/3)dx

Accepted Answer

Welcome back everyone. Find the value of the integral from 0 to pi divided by 3 of tangent of x divided by 2 dx. A ln of 3/4, bln of 2/3, CLN of 4/3, and the ln of 12. So for this problem we're going to begin with u substitution. Let's suppose that u is equal to x divided by 2. Let's remember that the next step is to find the derivative of u with respect to x, which is going to be the derivative of x divided by 2, and that's basically one half. So now we can show that dx is equal to DU multiplied by 2 or basically 2 DU. Our next step is to find the new limits of integration. The bottom one. is going to be U1 equals 0 divided by 2, which is 0, and the top 1, U2, is going to be equal to pi divided by 3. Divided by 2 witches, 5 divided by 6. So from here, we can rewrite our integral in terms of U and DU. We're integrating from 0 up to pi divided by 6. The function that we're integrating is tangent of u and dx becomes 2Du. We can write duu and we can factor out 2. Well then, from here we can integrate. We get 2, which is our constant. We have an integral of tangent of uDu. Let's remember the value that would be negative ln of the absolute value of cosine of eu. And now what we want to do is simply find the result from 0 up to pi divided by 6 by applying the fundamental theorem of calculus, part 2. We can also factor out the constant, so that would be -2. In parentheses, let's go ahead and evaluate. LM Of the absolute value of cosine of pi divided by 6. And we're going to subtract lm of the absolute value of cosine of zero. Now let's simplify using the properties of logarithms that would be -2 lm of the absolute value of. Cosine of pi divided by 6. Divided by cosine of 0. And now what we can do is simply replace cosine of pi divided by 6 and cosine of 0 with their actual values. So cosine of pi divided by 6 is going to be the square root of 3 divided by 2. And our cosine of 0 is equal to 1. Now we can drop the absolute value because LN of the square root of 3 divided by 2, well, essentially the square root of 3 divided by 2 is positive, right? So we don't have to use the absolute value. And from here we are going to simplify. Let's go ahead and do that. So, essentially, That would be ln of square root of 3 divided by 2 raise to the power of -2. We're applying the power rule of logarithms, right? We're going to use -2 as an exponent. This is equal to ln 2 divided by a square root of 3 squared using the reciprocal rule. And now finally we're going to get Ean of 4 divided by. 3 Considering the answer choices, we can conclude that the correct answer is option C. Thank you for watching.

Evaluate the integrals in Exercises 31–78.39. ∫(from 0 to π)tan(x/3)dx

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Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 to π)tan(x/3)dx