Determine which of the following two results is correct: ∫(x+1)^3 d x=(1)/(4)(x+1)^4+C or ∫(x+1)

step 1 So for this problem, what you can do to see which problem is correct is taking the derivative of

Determine which of the following two results is correct:...

Key Concepts

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by introducing a new variable, typically when the integrand contains a composite function. By letting u equal some function of x, the integral can often be converted into an easier form to integrate, as is seen with integrals of the form (x+1)^3.

Verification by Differentiation

Verification by differentiation is a method for checking the correctness of an antiderivative. After finding a potential antiderivative, differentiating it should yield the original integrand. This process is crucial to confirm that the applied integration method, whether through substitution or direct power integration, has been correctly executed.

Indefinite Integration

Indefinite integration is the process of finding an antiderivative or a family of functions whose derivative is the given integrand. It involves calculating an antiderivative and includes an arbitrary constant, since differentiation of a constant yields zero.

Power Rule for Integration

The power rule for integration is a fundamental technique used to integrate functions of the form u^n, where n is a real number not equal to -1. It states that the integral of u^n with respect to u is u^(n+1)/(n+1) plus a constant, and it is directly applied when the integrand is a simple power of a variable or shifted variable.

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