Evaluate the integrals in Exercises 1–22.
∫₀^π 3sin(x/3)...

Question

Evaluate the integrals in Exercises 1–22.

∫₀^π 3sin(x/3) dx

Accepted Answer

Welcome back everyone. Evaluate the definite integral from 0 up to 2 pi of 4 cosine of x divided by 4 dx. A 16, B8 C 0, and D 32. So for this problem we're going to use the u substitution. Let's suppose that u is equal to x divided by 4, and let's remember that our next step is to differentiate u with respect to x, which is the derivative of 1/4 x. That's equal to 1/4, and we can now show that dx multiplied by 1 is equal to duu multiplied by 4. So d x is equal to 4 duu. Which allows us to rewrite the integral in terms of u and DU. But before we do that, let's also change the limits of integration. So the lower 1 u1 is going to be. 0 divided by 4 based on the relationship between u and x. This gives us 0, and the upper limit of integration u2 is going to be 2 pi divided by 4. Which is equal to pi divided by 2. So now the integral can be written as the integral from 0 up to pi divided by 2 of 4, cosine of u multiplied by dx, which is 4DU. Now 4 multiplied by 4 is 16, and we can factor out 16 because it's a constant. And we can integrate. Cosine of uDu from 0 up to pi divided by 2. Let's remember that the integral of cosine is sin, so we get 16 sine of u evaluated from 0 up to pi divided by 2. And now we can calculate the result, factoring out 16 in parentheses, we will take sine of 5 divided by 2 and subtract sin of 0. sin of 0 is 0 and sine of pi divided by 2 is a 1, so we get 16 multiplied by 1 which is 16. And this corresponds to the answer of choice a. Thank you for watching.

Evaluate the integrals in Exercises 1–22.∫₀^π 3sin(x/3) dx

Watch next

Evaluate the integrals in Exercises 1–22.
∫₀^π 3sin(x/3) dx