Find the Antiderivative f(x)=ax^6

Problem

(∫_^)(a*x^6*d(x))

Solution

1. Identify the constant and the power of x in the expression a*x^6

2. Apply the power rule for integration, which states that (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)+C for any n≠−1

3. Add one to the exponent: $6 + 1 = 7$.

4. Divide the expression by the new exponent 7 and keep the constant a as a coefficient.

5. Include the constant of integration C to represent the family of all possible antiderivatives.

Final Answer

(∫_^)(a*x^6*d(x))=(a*x^7)/7+C

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