Find the Antiderivative f(x)=6(2x+1)^5(2)

Problem

(∫_^)(6*(2*x+1)5*(2)*d(x))

Solution

1. Simplify the integrand by multiplying the constant factors 6 and 2

(∫_^)(12*(2*x+1)5*d(x))

2. Identify the structure of the integral, which follows the power rule for a linear composition (∫_^)((a*x+b)n*d(x))

(∫_^)(12*(2*x+1)5*d(x))=12*(∫_^)((2*x+1)5*d(x))

3. Apply the reverse chain rule or usubstitution where u=2*x+1 and d(u)=2*d(x)

(∫_^)((2*x+1)5*d(x))=((2*x+1)6)/(6⋅2)

4. Multiply the constant 12 by the result of the integration.

12⋅((2*x+1)6)/12

5. Simplify the expression by canceling the 12 in the numerator and denominator and adding the constant of integration C

(2*x+1)6+C

Final Answer

(∫_^)(6*(2*x+1)5*(2)*d(x))=(2*x+1)6+C

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