Evaluate the Integral integral of 2(2x+4)^5 with respect to x

Problem

(∫_^)(2*(2*x+4)5*d(x))

Solution

1. Identify the integral as a power of a linear function, which suggests using usubstitution.

2. Substitute u=2*x+4

3. Differentiate u with respect to x to find d(u)=2*d(x)

4. Rewrite the integral in terms of u by replacing 2*x+4 with u and 2*d(x) with d(u)

(∫_^)(u5*d(u))

5. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u6)/6+C

6. Back-substitute the original expression 2*x+4 for u

((2*x+4)6)/6+C

Final Answer

(∫_^)(2*(2*x+4)5*d(x))=((2*x+4)6)/6+C

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