Variations on the substitution method Evaluate the following i...

Question

Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(∛𝓍 + 4) d𝓍

Accepted Answer

Welcome back, everyone. Evaluate the integral of XDX divided by cubic root of X + 7. For this problem, let's begin with the U substitution. Let's suppose that U equals x + 7. What we're going to do is notice that our numerator contains X. So let's also solve for X. X is equal to U minus 7. We also want to identify the relationship between DU and DX. So let's find the derivative DU divided by DX, which is the derivative of X + 7, and this is equal to 1. Therefore, the X is equal to DU. Now we can rewrite our integral in terms of UNDU. First of all, we have XDX and that would be U minus 7 multiplied by DU. Our denominator contains cubic root of X plus 7, which is cubic root of you, or you raise the power of 1/3. And now we can split our integral. We can divide the U minus 7 by U raise to the power of 1/3. We get you divided by you raise the power of 1/3, which is you raise the power of 2/3, the EU. That's our first integral, and then we subtract 7 integral DU divided by you raise the power of 1/3, which is you raise the power of -1/3 DU. And now let's go ahead and integrate each part using the power rule. For the first integral, we get you raise to the power of 5/3 divided by 5/3. And for the second integral, we have already factored out the constant. Let's apply the power rule. We get you raised to the power of 2/3 divided by 2/3, plus a constant of integration C. And now let's begin simplifying. For the first integral dividing by 5/3, we get 3/5. You raise the power of 5/3. Minus 7 multiplied by 3 divided by 2 is 21 halves. You raises the power of 2/3 plus a constant of integration C. And now let's go ahead and find the common denominator and factor. Our common denominator is going to be 10. Now, let's consider our fractions. So, 3/5 is equivalent to 6 divided by 10. We have 6 divided by 10, you raise the power of 5/3 minus 21 divided by 2 is the same as 105 divided by 10. You raise the power of 2/3 plus a constant of integration C. And now what we can do is simply factor out 3 divided by 10, considering the U terms we're going to factor out U raise to the power of. 2/3. And our ports we're going to have to you. Minus 105 divided by 3 is 35. And that's it, right? Because we have already factored out you raised the power of 2/3 plus a constant of integration C. And finally, we can replace you with X + 7, so we get 3 divided by 10. 2/3 essentially represents cubic root. Of u squad, so we get cubic root of X + 7 squad. In parents we have to you. Now to you is going to be 2 X + 14. And we subtract 35, so we got 2 X minus 21, because 14 minus 35 is 21 and we add a constant of integration C and that would be our final answer for this problem 3 divided by 10 multiplied by a cubic root of X plus. 77 squared multiplied by 2 X minus 21 plus a constant of integration C. Thank you for watching.

Variations on the substitution method Evaluate the following integrals. ∫ 𝓍/(∛𝓍 + 4) d𝓍

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Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(∛𝓍 + 4) d𝓍