Show that (i) ∫a^b lnx d x=b(lnb-1)-a(lna-1) for all a, b ∈(0, ∞), (ii) ∫a^b x d x=b-a for all a

step 1 In this video, we are going to be calculating the indefinite integral of x times b to the x d x.

Show that (i) ∫a^b lnx d x=b(lnb-1)-a(lna-1) for all a, b...

Key Concepts

Fundamental Theorem of Calculus

This theorem connects differentiation with integration by stating that if a function is continuous over an interval, then its definite integral over that interval can be computed using any of its antiderivatives. Specifically, by finding an antiderivative F of a function f, the definite integral from a to b is given by F(b) - F(a), a formula used in evaluating both integrals in the question.

Antiderivatives

An antiderivative is a function whose derivative is the given function. When computing definite integrals, finding an antiderivative is crucial because it allows one to apply the Fundamental Theorem of Calculus. For example, the integration of ln(x) and exp(x) requires knowledge of their respective antiderivatives.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It is particularly useful for integrating the natural logarithm function, ln(x), which does not have an immediate antiderivative in standard form. The rule helps transform the integral into a simpler form that can be evaluated more easily.

Properties of the Exponential Function

The exponential function, exp(x), is unique because its derivative is itself. This property simplifies its integration, as the antiderivative of exp(x) is directly exp(x) plus a constant. This self-replicating nature is the reason why the integral of exp(x) over any interval is simply the difference exp(b) - exp(a).

Transcript

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