Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ ...

Question

Evaluate the integrals in Exercises 39–54.

∫ 1 / (x⁶(x⁵ + 4)) dx

Accepted Answer

Welcome back everyone. Evaluate the integral 1 divided by x cubed x2 + 4 d x. For this problem we're going to use the use substitution, but before we begin, let's multiply both sides of the fraction by x, which essentially gives us integral of x. Divided by x the power of 4. Multiplied by x2 + 4, followed by dx. So what we're going to do from here is introduce a U substitution. In particular, we're going to set u equals x2. The derivative of u with respect to x is going to be 2 x, and we can now show that x multiplied by dx is d u divided by 2. So that 2 X multiplied by the X is DU. And therefore XDX is 1/2 the. Well done so from here. We can rewrite the integral as integral of x dx which becomes 1/2 d u. We can simply factor out one half, and we can write DU. In the numerator. In the denominator we have x to the power of 4, which is u squared. Followed by x2 + 4, and that's u + 4. And from here we can use partial fraction decomposition. So the function that we're integrating is 1 divided by u2 u + 4. It can be decomposed into a divided by u. Less. B divided by U2 + C divided by U + 4. Now from here, let's multiply both sides of this equation by u2 u +4, which gives us 1 equals a multiplied by u multiplied by u + 4. Plus b multiplied by u + 4. Plusc multiplied by u 2 and we will identify a b and c. To find B, specifically, we will start with step one. We're going to set U equals 0 because it will eliminate A and C, right? So we got 1 equals B multiplied by 0 + 4. Or basically 1 equals 4b. And therefore B is equal to 1 divided by 4. Now we will find C and we will set u equals -4. Which will eliminate A and B, right? So we get one on the left hand side equals. C multiplied by -4 squared. C multiplied by -4 2, which gives us 1 equals. 16 c and therefore c equals 1 divided by 16. As a reminder to find B, we. Set U equals 0. And to find c, we set u equals -4. And now, let's find a. To find A, we basically have to notice that. 1 equals from our equation. What we're going to do is basically distribute, right? So we will get A U 2 + 4 A U +. BU + 4 B + CU 2. And specifically, we have to notice that A + C multiplied by U2, we can basically factor out these terms, right? So the coefficient in front of u 2 is 0 in the numerator. There's no necessity to group the remaining terms. We just have to notice that A + C equals 0. And therefore A is going to be negative C. Which is -1 divided by 16. Well done. So now we can rewrite the integral that would be 1/2 integral of a divided by u. So that's -1/16 divided by u or basically multiplied by 1 divided by u. Plus B. Which is 1/4, so we get plus 1/4 divided by u2 or multiplied by 1 divided by u2. Plus C. Which is -1 divided by 16. Divided by U + 4 or multiplied by 1 divided by u + 4. And we're integrating with respect to you. From here we can integrate. And before we integrate, we can also factor out. 1/16. In particular, it might be more useful to factor out -116. Let's consider our terms, right? And let's notice that. For the final term. We actually have 116, right? So let's correct that. Now From here let's factor out 1/16 instead, which will give us 1 divided by 32 in front of the integral. And then we're going to integrate. 1 divided by you. Plus. For you to the power of -2. Plus 1 divided by U + 4 DU. And now let's consider each part within this integral. So we get 1 divided by 32. The integral of -1 divided by you is. Negative lm of the absolute value of u. And now let's remember that U is x 2, so we can drop the absolute value. Plus 4. Integral of u to the power of -2 is u to the power of -1. Divided by a -1 using the power rule. Plus the integral of 1 divided by u + 4 is ln of u + 4. Plus a constant of integration c. And now what we're going to do is simplify. So for the middle term, plus for you to the power of -1 divided by -1, we can replace the positive sign with the negative sign. And right 4 divided by you. And let's apply the properties of logarithms. We get 1 divided by 32. In parenthes. LM of U + 4. Divided by you. Minus y divided by u plus c. Now let's go ahead and distribute, which gives us 1 divided by 32 ln of U plus 4 divided by U. Minus 1/8 Divided by you or 1 divided by 8U plus C. And finally, let's replace you with x 2, right, which is going to give us. Our final answer for this problem. Let's label it and thank you for watching.

Evaluate the integrals in Exercises 39–54.∫ 1 / (x⁶(x⁵ + 4)) dx

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Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx