2–9. Integrals Evaluate the following integrals.

Question

∫₁⁴ (10^{√x} / √x) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral from 1 to 16 of 2, raise to the power of square root of X, divided by square root of X, multiplied by DX. For this problem we're going to use the U substitution. Let's suppose that U is equal to square root of X. Our next step is to evaluate the derivative of u with respect to X, which is the derivative of square root of X. We know that this is equal to 1 divided by 2 square root of x. Now, what we can do from here is simply understand that U is equal to square root of X, right? So this is equivalent to 1 divided by 2 U. Now we're going to solve for DX. DX is going to be equal to DU divided by. 1 divided by 2 U. so we're dividing DU by the right hand side, and this is equal to 2U multiplied by DU. We are now going to identify the lower limit of integration U1, which is square root of 1, and that's 1, and the upper limit of integration U2 is square root of 16, which is equal to 4. So here we're using the relationship between you and X, right? U equals square root of X. So we're taking every limit and performing the calculation square root of That limit square root of 1 and square root of 16. We can now rewrite our integral as the integral from 1 to 4. Of 2 square root of, to raise the power of square root of X. So that would be to raise the power of you. We are now dividing by square root of X, which is you. And multiplying by DX. DX is to UDU. Now notice how we can cancel out you. And this gives us The integral from 1 to 4. Of 2 multiplied by 2 raises the power of you. The EU we can factor out the constant. We get 2. Integral from 1 to 4 of 2 raised the power of UDU. And then we can integrate. We get 2, which is our constant multiplied by. The integral of 2 raise the power of UDU can be evaluated using the tables, so that'd be 2 raise the power of U divided by LN of 2. And we are now evaluating the result from 1 to 4. Let's perform the calculation. We can factor out our constant, which is 2 divided by LN of 2, and in paris we can evaluate to raise the power of 4 minus 2 raise the power of 1. This is equal to 2 divided by Llan of 2 multiplied by. 2 raises the power of 416 and we're subtracting 2, so we're multiplying by 14 and 14 multiplied by 228. So we get 28 divided by LN of 2, which is our final answer for this problem. Thank you for watching.

2–9. Integrals Evaluate the following integrals.

∫₁⁴ (10^{√x} / √x) dx

Key Concepts

Definite Integrals

Substitution Method

Exponential Functions with Variable Exponents

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