Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t...

Question

Evaluate the integrals in Exercises 39–54.

∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

Accepted Answer

Welcome back everyone. Evaluate the the integral e to the power of 4 x plus 3 e to the power of 2 x plus 2 dx divided by eats the power of 2 x plus 3. For this problem, let's begin by using u substitution and let's suppose that u is equal to e to the power of 2 x. Then the derivative of u with respect to x according to the chain rule is going to be e to the power of 2 x multiplied by 2 x derivative, which is 2. We can now solve for DX and DX is going to be DU. Divided by 2 e. To the power of 2 X, which is basically DU divided by 2 U, right? Because E to the power of 2 X is U. What we can now do is rewrite the integral in terms of UNDU. So in the numerator we have u2 + 3 u + 2. In the denominator, we have U + 3. And now DX becomes DU divided by 2 U. What we can do is factor out. 1/2 to get 1/22 integral. Of E U 2 + 3, U + 2, followed by DU. And we can use. The u term in the denominator and multiply it by u plus 3 in the other denominator. So that's going to be U 2 + 3EU. Well done. And now let's go ahead and divide. So that's going to be 1/2 integral. U2 + 3U divided by itself is going to give us 1. And we're left with + 2 divided by U2 + 3U Followed by the So now we can integrate 1DU easily. Let's focus on. The next integral, which is 2 divided by u2 + 3 u. We're going to rewrite it, 2 divided by U2 + 3U, and we're going to factor the denominator, that's 2 divided by U, U plus 3. What we can do is use partial fractions, so that's going to be a divided by U + B divided by the second linear factor U + 3. And now let's go ahead and identify the constants A and B. We can multiply both sides by u u plus 3 to get 2 equals a. Multiplied by u + 3. Plus B multiplied by you. First of all, we're going to identify a and to find a, let's set u equals 0, which will eliminate the b term. If you equals 0, we get 2 equals a multiplied by 0 + 3 or simply 3a. So A is going to be 2 divided by 3. From here we will identify. B To find B, let's set u equals -3. Which gives us 2 equals B multiplied by -3 or -3B. So B is going to be 2/3. Well then, and now let's identify the overall integral i. We're going to label the integral with I for simplicity. And we can show that it is equal to 1/22. Integral Of One Plus Now, the expression that we have can now be decomposed into two fractions, so we get a, which is 2/3. Divided by you or simply multiplied by one divided by you, right? Plus b, which is 2/3 divided by u +3 or multiplied by 1 divided by u + 3. And we are integrating with respect to you. Now we can split into 3 integrals. And factor out each constant which gives us one half integral of d u. Plus 1/2 multiplied by 2/3 is going to be 2 divided by 6 or simply 1/3. Integral Of 1 divided by UDU. My nurse 1/3 We get it by multiplying 1/2 by 2/3, right? Integral 1 divided by u plus 3DU. And now these are elementary integrals, so for the first one we get 1/2. You, that's the integral of the you. Plus 1/3 Multiplied by The integral of 1 divided by u d u that's ln of the absolute value of u minus 1/3. The integral of 1 divided by u + 3 d u is ln of the absolute value of u + 3. And we are going to add a constant of integration C. Finally, we are going to replace you with E to the power of 2 x, right? And we are going to simplify. So let's go ahead and do that. We get one half U, which is e to the power of 2 X. Plus 1/3 ln of the absolute value of e to the power of 2 x. E to the power of 2 x is always greater than 0, so we can drop the absolute value, and we can also apply the properties of logarithms, right? So ln of e to the power of 2 x is simply 2 x. So we're going to get 2 eggs divided by 3. Or 2/3 X. Minus 1/3 lm of the absolute value of e to the power of 2 x + 3. Again we can drop the absolute value because e to the power of 2 x + 3 is going to be greater than 0. Bless a constancy. That's our final answer and thank you for watching.

Evaluate the integrals in Exercises 39–54.∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

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Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt