Evaluate the Integral integral of 4(4x+3)^4 with respect to x

Problem

(∫_^)(4*(4*x+3)4*d(x))

Solution

1. Identify the structure of the integral, which is in the form of a power of a linear function.

2. Apply the substitution method by letting u=4*x+3

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

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

(∫_^)(u4*d(u))

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

(u5)/5+C

6. Back-substitute the original expression 4*x+3 for u to get the final result.

Final Answer

(∫_^)(4*(4*x+3)4*d(x))=((4*x+3)5)/5+C

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