Evaluate the Integral integral of (x^4+9)^8 with respect to x

Problem

(∫_^)((x^4+9)^8*d(x))

Solution

1. Expand the binomial expression (x^4+9)^8 using the Binomial Theorem. The general formula for expansion is (a+b)^n=(∑_k=0^n)((n/k))*a^(n−k)*b^k)

2. Identify the terms of the expansion where a=x^4 b=9 and n=8

3. Write out the full polynomial:

(8/0)(x^4)^8*(9)^0+(8/1)(x^4)^7*(9)^1+(8/2)(x^4)^6*(9)^2+(8/3)(x^4)^5*(9)^3+(8/4)(x^4)^4*(9)^4+(8/5)(x^4)^3*(9)^5+(8/6)(x^4)^2*(9)^6+(8/7)(x^4)^1*(9)^7+(8/8)(x^4)^0*(9)^8)))))))))

4. Simplify the coefficients and powers of x

x^32+72*x^28+2268*x^24+40824*x^20+459270*x^16+3306744*x^12+14880348*x^8+38263752*x^4+43046721

5. Integrate each term individually using the power rule (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)

6. Add the constant of integration C

Final Answer

(∫_^)((x^4+9)^8*d(x))=(x^33)/33+(72*x^29)/29+(2268*x^25)/25+(40824*x^21)/21+(459270*x^17)/17+(3306744*x^13)/13+(14880348*x^9)/9+(38263752*x^5)/5+43046721*x+C

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