64. Using a computer algebra system, it was determined that

Question

∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.

Use integration by substitution to evaluate ∫x(x+1)^8 dx.

Accepted Answer

Welcome back, everyone. The following integral is obtained using a computer algebra system. Integral of X multiplied by X + 1 to the power of 9DX is equal to X to the power of 11 divided by 11 + 9 X to the power of 10 divided by 10 + 4x the power of 9, plus 21 X to the power of 8. Divided by 2 + 18 X to the power of 7, plus 21 X raises the power of 6, plus 84 X the power of 5 divided by 5, plus 9 X the power of 4, plus 3X cubed plus 1/2 X2 plus C. Evaluate this integrally using integration by substitution. Let's go ahead and introduce substitution U equals X + 1. Since we have X in the integrant, let's also solve for X. X is equal to U minus 1. And now we can use Our you substitution. To evaluate DU, right? So let's identify the derivative DU divided by DX, which is the derivative of X + 1, and that's 1, meaning the U is equal to DX, right? So now we can rewrite our integral as X, which is U minus 1. Multiplied by x + 1 which is u erase the power of nine and LDX becomes D U. Now, let's go ahead and distribute. We're going to integrate. You raise the power of 10 minus U, raise the power of 9DU. We can integrate using the power rule. We get you raise the power of 11 divided by E11 minus U raises the power of 10 divided by 10 plus e constant of integration C because this is an indefinite integral. Before we substitute back, let's identify the common denominator, which is 110. So the first term becomes 10, you raise the power of 11 divided by 1010, and for the second sum we're subtracting 11, you raise the power of 10 divided by 100 and 10. We're adding a constant of integration, see. And now we can perform factorization, right? So let's go ahead and factor out one divided by. 110 and imprecies we have 10 you raise the power of 11 minus. 11, you raise the power of 10 plus a constant of integration C. Now we can notice that we can also factor out you raise the power of 10. So in priss we have 10 U minus 11. And then we can substitute you back in, so we got X + 1, raise to the power of 10. Divided by 110 and we have 10 X plus 1 minus 11 plus C. And now let's distribute and combine alike terms. We can state our answer as one divided by 110 multiplied by X plus 1 raise the power of 10. In pres we get 10 X. Plus 10 minus 11, so that's -1 plus a constant of integration C, and that would be our final answer for this problem. Thank you for watching.

64. Using a computer algebra system, it was determined that∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.Use integration by substitution to evaluate ∫x(x+1)^8 dx.

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64. Using a computer algebra system, it was determined that
∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.
Use integration by substitution to evaluate ∫x(x+1)^8 dx.