In Exercises 43-46, verify by differentiation. ∫(a x+b)^n d x=(1)/(a(n+1))(a x+b)^n+1+C (

step 1 This is problem number 46. 46 we have to verify by differentiation that a x plus b raised to the

In Exercises 43-46, verify by differentiation. ∫(a x+b)^n d...

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Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental method for integrating functions raised to a power, provided that the exponent is not -1. It determines that the antiderivative of a function of the form u^n is given by u^(n+1)/(n+1), where u is a function of x, and adjustments are made for any constant multipliers associated with the transformation.

Verification by Differentiation

This concept involves confirming that an antiderivative is correct by differentiating it. It relies on the Fundamental Theorem of Calculus, which states that differentiation is the inverse process of integration. By differentiating the proposed antiderivative and obtaining the original integrand, one verifies the integration result.

Chain Rule

The chain rule is a key differentiation technique used when dealing with composite functions. In this context, when differentiating a function of the form (ax + b)^(n+1), the chain rule ensures that the derivative of the inner function (ax + b) is correctly accounted for, which provides the necessary constant factor in the derivative.

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