Find the Integral (x^4+9)^8(4x^3dx)

Problem

(∫_^)((x4+9)8*(4*x3)*d(x))

Solution

1. Identify the structure of the integrand and notice that the term 4*x3 is the derivative of the inner function x4+9

2. Apply substitution by letting u=x4+9

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=4*x3*d(x)

4. Substitute u and d(u) into the integral to rewrite it in terms of u

(∫_^)(u8*d(u))

5. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u9)/9+C

6. Back-substitute the original expression for u to get the final result in terms of x

Final Answer

(∫_^)((x4+9)8*(4*x3)*d(x))=((x4+9)9)/9+C

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