Evaluate the Integral integral of (2x+6)^5 with respect to x

Problem

(∫_^)((2*x+6)5*d(x))

Solution

1. Identify the inner function for substitution. Let u=2*x+6

2. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=2

3. Solve for d(x) to substitute it into the integral.

d(x)=d(u)/2

4. Substitute u and d(x) into the original integral.

(∫_^)(u5⋅1/2*d(u))

5. Factor out the constant from the integral.

1/2*(∫_^)(u5*d(u))

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

1/2⋅(u6)/6+C

7. Simplify the expression.

(u6)/12+C

8. Back-substitute the original expression for u which was 2*x+6

((2*x+6)6)/12+C

Final Answer

(∫_^)((2*x+6)5*d(x))=((2*x+6)6)/12+C

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