2–74. Integration techniques Use the methods introduced in Sec...

Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

Accepted Answer

Hello. In this video, we are going to be evaluating the following integral. So we are given the integral from 1 to 2 of DX divided by 2 X 2 plus 8 X plus 9. Now, what we want to do in this problem is you want to simplify the denominator in such a way that we can approach this integral by using inverse trigonometric identities. So, in the denominator, we have 2 X2d. Plus 8 X plus 9. We can start by factoring out a 2 from the first two terms. That will give us 2 multiplied by X2 + 4 X plus 9. Then if we were to complete the square on X2 + 4 X, we will be left with 2 multiplied by X2 + 4 X + 4. Then to balance the equation, we must subtract a 4 multiplied by 2. This is going to simplify to be 2 multiplied by X + 2 quantity squared. And 9 minus 4 multiplied by 2 is going to leave us with positive 1. So we can rewrite the integral to be the integral from 1 to 2 of DX divided by 2 multiplied by X + 2 quantity squared plus 1. Next, what we can do is we can simplify this integral further by using a U substitution. We will allow you to equal to X + 2. Then we need to come up with a substitution for DX. So by taking the derivative of the U substitution in terms of X, we will have DU is equal to DX. By applying these substitutions, we will have the integral of 1 divided by 2 Us squared plus 1 DU. Then we need to change our bounds of integration. We will take our bounds of integration and plug them into the new substitution, and our new bounds of integration are going to be from 3 to 4. Next, what we can do is we can factor out a 2 from the denominator and move that as a constant in front of the integral. That is going to give us 1/2 multiplied by the integral from 3 to 4 of 1 divided by U2 plus 1/2 DU. Now, we can substitute, or we can simplify this integral by using the tangent inverse integral. If we have an integral in the form of x2 plus a squad. Then this is equal to 1 divided by A, multiplied by tangent inverse of X divided by A. Here, our A square term is equal to 1/2, and that means our A term is going to be the square root of 1/2. And this will further simplify to be 1 divided by the square root of 2, which will rationalize to be square root 2 divided by 2. So, by using this identity on the current integral, we can simplify the integral to be one half, multiplied by the quantity, square root 2 divided by 2, multiplied by tangent inverse. Oh, apologies. It is 1 divided by 8. So that is going to be 1 divided by square root 2 divided by 2, multiplied by the tangent inverse of X divided by the square root 2 divided by 2. And this is going to be evaluated from 3 to 4. By further simplifying, we will simplify this integran to be. The square root of 2, divided by 2 multiplied by the tangent inverse of the square root to U. And this will be evaluated from 3 to 4. Now what we need to do is we need to plug in the values of the bounds. By doing so, we get square root 2 divided by 2, multiplied by the tangent inverse of 4 square root of 2. Minus square root 2 divided by 2, multiplied by the tangent inverse. Of 3 square root of 2. Now, we can further combine the tangent inverse functions by using the following identity. If we have a difference. Of tangent inverse functions. Let's say tangent inverse of A minus the tangent inverse of B, then this simplifies to A minus B divided by 1. Plus A multiplied by B, and this will all be within the tangent inverse function. So, this is going to further simplify to be the square root of 2 divided by 2, multiplied by the tangent inverse of 4 square root of 2 minus 3 square root of 2. Divided by 1 minus. 4 square root of 2 multiplied by 3 square root of 2. And this is going to further simplify to give us the square root 2 divided by 2, multiplied by the tangent inverse of square root of 2 divided by 25, and this is going to be the solution to the given definite integral. And with that being said, the overall solution to this problem is going to be B. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

Key Concepts

Integration Techniques

Definite Integrals

Quadratic Functions

Watch next