In Exercises 41-44, verify by differentiation. ∫(a x+b)^n d x=(1)/(a(n+1))(a x+b)^n+1+C ( for

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In Exercises 41-44, verify by differentiation. ∫(a x+b)^n d...

Key Concepts

Integration of Power Functions

This concept refers to finding antiderivatives of functions that are raised to a power. The formula ?(ax+b)^n dx expresses the result as another power function with an adjustment for the coefficient and exponent. It encapsulates the idea that integration effectively reverses the differentiation process for these functions under the appropriate conditions (n ? -1).

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. In verifying the integration formula by differentiation, the chain rule is applied to differentiate the outer function raised to a power and then multiply by the derivative of its inner linear function. This ensures the correct recovery of the original integrand.

Power Rule for Differentiation

The power rule states that the derivative of a function in the form (f(x))^(n+1) is (n+1) times (f(x))^n multiplied by the derivative of f(x). This rule directly applies to the verification process, where differentiating (ax+b)^(n+1) yields the factor (n+1) and, through the chain rule, nicely cancels out with the constant coefficient in the integration formula.

Verification by Differentiation

Verification by differentiation is the process of confirming an antiderivative by differentiating it to see if the original function is obtained. This technique ensures that the computed antiderivative is correct, adhering to the Fundamental Theorem of Calculus. It is an essential step in calculus to validate integration results properly.

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