Evaluate the integrals in Exercises 97–110.
105. ∫₀² (lo...

Question

Evaluate the integrals in Exercises 97–110.

105. ∫₀² (log₂(x + 2) / (x + 2)) dx

Accepted Answer

Welcome back everyone. Evaluate the the integral from -1 up to 1 off log of base 2 of 4 x + 5. Dx divided by 4 x plus 5. A LN +23 divided by 2 LN2. BLN +23 divided by 8 LN2. CLN 3 divided by 22, and DLN 3 divided by 8 LN 2. For this problem we're going to begin with changing the base of the logarithm because normally we work with natural logarithms. So log of base 2 of 4 x plus 5 in terms of ln would be ln of 4 x. Plus 5 divided by ln of the original base, so that'd be ln of 2. And therefore we can rewrite the integral as follows, that'd be the integral from -1 up to 1. In the numerator we have ln of 4 x plus 5 divided by ln of 2. And then we're dividing by 4 x plus 5. So basically what we can do is write the denominator as LN of 2. Multiplied by 4 x plus 5, this is our combined denominator since we have that double division. From here we can factor out the constant. It is 1 divided by ln of 2. And the integral is from -1 up to 1 of L and. Of 4 x plus 5 divided by 4 x plus 5, let's also add dx, right, to show that we're integrating with respect to x. Now what we can do is introduce U substitution. Let's suppose that U is equal to LN of 4 x + 5, and let's remember that the next step is to find the derivative of U with respect to x. That's the derivative of ln of 4 x + 5, and it is a composite function. So first of all we differentiate the natural log with respect to the inner function which gives us 1 divided by the inner function or 1 divided by 4 x plus 5, and according to the chain rule we're multiplying by the derivative of 4 x plus 5, which is 4. So overall we get. 4 divided by 4 x plus 5. And I'll notice that. Within the integrant we have d x divided by 4 x plus 5, so we're going to rearrange this expression as follows. Multiplying both sides by DX and dividing both sides by 4, we get DU divided by 4. Or basically 1/4 duu is equal to dx. Divided by 4 x plus 5. Well done, and now let's change the limits of integration, so the lower one is going to be ln of 4 multiplied by -1 + 5. Which is ln of 1. And that's basically 0, right? And the upper one is ln of 4 multiplied by 1 + 5, and that's. Alan of 9 Let's remember that L of 9 can be written as ln of 3 squad, which allows us to apply the properties of logarithms we can bring down the exponent, and that would be 2 ln of 3. So now from here, we can rewrite the integral in terms of UNDU. We're integrating from 0 up to 2A of 3. We have our constant, which is 1 divided by ln of 2. Now dx divided by 4 x plus 5 becomes 1/4 duu, so we can just write DU and we can factor out 1/4. Which gives us a combined constant of 1 divided by 4 ln of 2. And ln of 4 x + 5 basically becomes our u. Well done, so now we have an elementary integral we can integrate using the power rule to get our constant, which is 1 divided by 4 LN of 2. Multiplied by U 2 divided by 2 using the power rule, U to the power of 1 + 1 divided by 1 + 1, right? And we went to evaluate the result from 0 up to 2LN of 3. So now let's find the, the new combined constant, which is going to be 1 divided by 4 multiplied by 2, gives us 8. So that's 1 divided by 8 LN 2. And in parentheses, we're going to evaluate E U 2. At To a land of 3, right? So that'd be. 22 LN 2 of 3. And we're going to subtract the value at 0, so you squared at 0. Is basically 02, which is 0. And now what we want to do is simplify, so we can cross out 02, it's 0. 22 is 4. Right, so we can just continue the equation and show that it is equal to 4 lm square of 3. Divided by 8 LN of 2. And dividing both sides by 4, we get lm squared off 3. Divided by 2 LN of 2. And now let's consider the answer choices. This answer corresponds to the answer choice A. Thank you for watching.

Evaluate the integrals in Exercises 97–110.105. ∫₀² (log₂(x + 2) / (x + 2)) dx

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Evaluate the integrals in Exercises 97–110.
105. ∫₀² (log₂(x + 2) / (x + 2)) dx