Evaluate the integrals in Exercises 33–54.
∫(from ln4 to...

Question

Evaluate the integrals in Exercises 33–54.

∫(from ln4 to ln9)e^(x/2)dx

Accepted Answer

Welcome back everyone. Evaluate the integral from ln of 9 up to ln of 16 of e to the power of x divided by 2 d x. A 3, B2, C1 and D14. We're going to approach this problem using u substitution. Let's suppose that U is equal to x divided by 2, and let's remember that our next step is to differentiate u with respect to x. That's the derivative of 1/2 x, which is 1/2. So now we can show that dx multiplied by 1 is equal to duu multiplied by 2, meaning dx is equal to 2Du. Our next step is to change the limits of integration, so the lower one is going to be x divided by 2, that's 1/2 ln of 9. Which can be simplified using the properties of logarithms, that's ln of 9 to 1/2, and this is equal to ln of square root of 9, which is ln of 3. And the upper limit of integration is going to be 1/2 ln of 16. Which is a land of 16 to the power of 1/2, and that's the land of 16. A land of square root of 16, which is alan of 4. So the integral becomes integral from LN of 3 up to LN 4. Of e to the power of u and dx becomes 2 d u. So we can write duu and we can factor out 2 because 2 is a constant. So now we're going to get 2, and we have an elementary integral. The integral of e to the power of u is e to the power of u, and we want to evaluate the result from LN of 3 up to LN of 4 using the fundamental theorem of calculus part 2, which gives us 2. In parentheses is the power of e of 4 minus e the power of ean of 3. This is going to be 24 minus 3, which is equal to 2 multiplied by 1, and that's 2. Meaning the correct answer is option B. Thank you for watching.

Evaluate the integrals in Exercises 33–54.∫(from ln4 to ln9)e^(x/2)dx

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Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx