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

Problem

(∫_^)((x4+9)8*d(x))

Solution

1. Expand the binomial expression (x4+9)8 using the Binomial Theorem. The general formula for expansion is (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

2. Identify the terms of the expansion where a=x4 b=9 and n=8

3. Write out the full polynomial:

(8/0)(x4)8*(9)0+(8/1)(x4)7*(9)1+(8/2)(x4)6*(9)2+(8/3)(x4)5*(9)3+(8/4)(x4)4*(9)4+(8/5)(x4)3*(9)5+(8/6)(x4)2*(9)6+(8/7)(x4)1*(9)7+(8/8)(x4)0*(9)8)))))))))

4. Simplify the coefficients and powers of x

x32+72*x28+2268*x24+40824*x20+459270*x16+3306744*x12+14880348*x8+38263752*x4+43046721

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

6. Add the constant of integration C

Final Answer

(∫_^)((x4+9)8*d(x))=(x33)/33+(72*x29)/29+(2268*x25)/25+(40824*x21)/21+(459270*x17)/17+(3306744*x13)/13+(14880348*x9)/9+(38263752*x5)/5+43046721*x+C

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