Evaluate the integrals in Exercises 9–28. It may be necessary ...

Question

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [1 / (x (1 + ∛x))] dx

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral, the integral of 3 divided by x multiplied by the parentheses of 8 plus the cube root of x dx. OK, so it appears for this particular prompt for us to simply evaluate the integral that has been provided to us. So now we know that we're ultimately trying to evaluate this integral. We need to recall a note based on our prior knowledge in order to evaluate the integral of 3 divided by x multiplied by the parentheses of. 8 plus the cube root. Of X. DX we will need to use a substitution method in order to eliminate the cube root radical, which will therefore turn the expression into a rational function. That we can solve for using partial fractions. So keeping that in mind, our first step is we need to use our substitution method. So let us note that you for this particular prompt will be equal to the cube root of x. And x will be equal to u to the power of 3, and that will mean that dx by DU will be equal to 3 multiplied by u to the power of 2, and dx will be equal to 3 multiplied by u to the power of 2 duu. And we need to substitute these variables into our integral, and when we do, we will be able to write that the interval of 3 divided by u to the power of 3 multiplied by parentheses of 8 plus u. Multiplied by 3 multiplied by u to the power of 2 DU. And we need to now simplify our expression by canceling u to the power of 2 from the numerator and denominator, and when we do that, we will get 9 multiplied by the interval. Of 1 divided by you multiplied by parentheses of you plus 8. Do you And now at this stage for our second step, we need to use partial fraction decomposition, so we will need to decompose the integrand into simpler fractions. So we need to take 1 divided by u multiplied by parentheses of u plus 8, which is equal to a divided by U plus B divided by U plus 8. Which will mean now we need to multiply both sides by the common denominator, which is, and let's make a note that our common denominator will be U multiplied by parentheses of U + 8. And when we once again multiply both sides by our common denominator, we will be able to write that 1 is equal to a multiplied by parentheses of U plus 8 plus B multiplied by U. Fantastic. So, we now need to solve for A, and in order to solve for A, we need to let U be equal to 0, and when we do that, we will find that 1 minus A multiplied by 0 plus 8, and when we rearrange this expression to isolate and solve for A, we will find that A is simply equal to 1/8. And to solve for B, we need to let U be equal to -8, and solving for B, similarly, we need to note that 1. Will be equal to, so 1 is equal to B multiplied by -8, which will mean that B is equal to -18, fantastic. So moving right along here. So that will mean we now need to substitute our constants back into our integral. So we need to take 9 multiplied by the integral of parentheses of 1/8 divided by you, and we need to take 1/8 divided by U minus 1/8 divided by U plus 8. Do you Which will be equal to drum roll 9 multiplied by the integral of parentheses of 1 divided by 8. You -1 divided by 8 multiplied by parentheses of U plus 8. Do you fantastic. Which will be all equal to 9 divided by 8 multiplied by the integral of parentheses of 1 divided by u minus 1 divided by u plus 8. Do you Fantastic. So now, for our third step is we need to integrate with respect to you. So you take 9 divided by 8, multiplied by the integral of parentheses of 1 divided by U minus 1 divided by you plus 8. And don't forget the u, which will be equal to drum roll, 9 divided by 8 multiplied by parentheses of the natural log of the absolute value of u minus the natural log of the absolute value of u plus 8. Plus C where c denotes some constant variable. So now we need to recall and use the logarithm property, which is the natural log of the absolute value of a minus the natural log of the absolute value of B will be equal to the natural log of. The absolute value of A divided by B. So when we apply this property, we will be able to go ahead and write that 9 divided by 8 multiplied by parentheses of the natural log of the absolute value of u minus the natural log of the absolute value of u plus 8. Plus C, which will be simply equal to. 9 divided by 8 multiplied by the natural log of U divided by U plus 8 and absolute value signs plus C. And now for our fourth and final step is we need to apply our back substitution. We need to note that U is equal to the cube root of x, which will mean that our final answer without a doubt will be 9 divided by 8 multiplied by the natural log of the absolute value of. The cube root of X divided by the cube root of X + 8. Plus C, and that is our final answer. Hooray, we did it. So that will mean looking at the problem itself. Our final answer once again for our evaluated integral is once again. 9 divided by 8 multiplied by the natural log of the absolute value of the cube root of x divided by the cube root of x plus 8 plus C. and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [1 / (x (1 + ∛x))] dx

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Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / (x (1 + ∛x))] dx