7–40. Table look-up integrals Use a table of integrals to eval...

Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the indefinite integral, and we're given the integral of E to the 4 Y divided by the square root of the quantity of 1 plus E to the 2Y and quantity D Y. So, in order to evaluate this integral, we'll need to use U substitution. So, we'll set U equal to 1 + E to the 2 Y. And so that means that DU is going to be equal to 2 multiplied by E to the 2 Y D Y. And so rewriting our integral, we'll have that the integral of E to the 4Y, but we're gonna rewrite E4Y as E to the 2Y, multiplied by E2Y. And that's gonna be divided by the square root of the quantity of 1 plus E to the 2 Y. In quantity, DY. So this is going to be equal to the integral of, and here we want to write e2Y in terms of you, and we can do that using our expression for you. So solving our equation for you for e2 Y, we see that E to these 2 Y is equal to U minus 1. So, in our interval, we'll have the quantity of U minus 1 in quantity, and then that gets multiplied by E 2 Y D Y, which in our case, is going to be equal to 12 multiplied by DU. Since DU is equal to 2 multiplied by E 2 Y, that means that 1/2 DU is equal to E2 Y D Y. And then this quantity is going to be divided by the square root of U. So what we want to do now is to rewrite our integrate here as two separate fractions. So doing so, this is going to be equal to the interval. Of 1/2 multiplied by the quantity of and for the first term we'll have you divided by the square root of you, which is going to be the square root of you, which we'll rewrite as u to the 12 power. Then we'll have minus 1 divided by the square root of you, which is you raised to the minus 1/2 in quantity, D U. So now we need to just integrate each term inside the integrat individually. So this is going to be equal to, we'll factor out the 1/2 in front. So we'll have 1/2 multiplied by the quantity, and when we integrate you to the 12 power with respect to you, we get you to the 3/2 power divided by the quantity of 3 divided by 2. In quantity. Then we'll have minus, and here we're integrating you to the -1 half power, and when we integrate you to the minus 1 half power, with respect to you, we get you to the 1 half power, that's gonna be divided by. The quantity of 1 divided by 2 in quantity. And then we'll have to add to this our integration constant for each of those integrations, so we'll just have plus C at the end for our total integration constant. So now we just need to simplify our expression here, as well as substitute in 1 + E to the 2Y for you. So this is going to be equal to one to have 1/2 divided by the quantity of 3 divided by 2. I get 1 divided by 3, and this gets multiplied by the quantity of 1 + E to the 2Y in quantity raised to the 3/2 power. Then we'll have minus for the next term, the 1/2s cancel, so we'll just have you to the 12 power. So that's just going to be the quantity of 1 + E to the 2 Y. In quantity to the one half power. Plus C. So, what we can do is factor out a quantity of 1 + E to the 2 Y in quantity to the 1/2 divided by 3 out of each of these terms. So doing so, this is going to be equal to. The quantity of 1 + E to the 2 Y and quantity to the 12 power. Divided by 3, multiplied by the quantity of 1, plus E to the 2 Y minus 3 in quantity, plus C. And finally, just simplifying our expression here, we see that this is going to be equal to the quantity of 1 plus e to the 2Y in quantity to the 12 power multiplied by the quantity of E2Y minus 2 in quantity, divided by 3 plus C. So this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

Key Concepts

Indefinite Integrals

Table of Integrals

Completing the Square

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

Key Concepts

Indefinite Integrals

Table of Integrals

Completing the Square

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt