Evaluate the integrals in Exercises 33–54.
51. ∫ from ln...

Question

Evaluate the integrals in Exercises 33–54.

51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv

Accepted Answer

Welcome back everyone. Evaluate the integral from 0 to LN of 3 of 4 e to the power of x, sin of e to the power of xDX. For this problem we're going to use substitution. Let's suppose that U is equal to e to the power of x, and let's remember that. Our next step is to find the derivative of u with respect to x. That's the derivative of e to the power of x, which is e to the power of x. If we multiply both sides by dx, we can show that d u is equal to e to the power of x d x. From here we're going to change the limits of integration. The lower 1 u1 is going to be e to the power of 0, which is 1, and the upper limit of integration u2 is e to the power of ln of 3, which is 3. So the integral can be rewritten in terms of u and d u. That's the integral from 1 up to 3 of. We can factor out 4 because it's a constant. Now e to the power of x becomes u. So essentially we are going to have u multiplied by. Sign Of u d x. However, we know that e to the power of x d x is du u. So instead of writing it in this form, right, we are going to use e to the power of x multiplied by dx, and we are going to combine it into d u. So the integral would be. 4 integral from 1 up to 3 of sin of u multiplied by e to the power of x dx which is d u and this shows us that we have an elementary integral. The integral of sine is negative cosine, so we get -4 cosine of u evaluated from 1 up to 3. Applying the fundamental theorem of calculus part two, we get -4 cosine. Of 3 minus -4 cosine of 1. Simplifying We get 4 cosine of 1 minus 4 cosine of 3, and additionally we can factor out 4 to get 4 in parentheses cosine of 1 minus cosine of 3. Which is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 33–54.51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv

Watch next

Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv