Show that ∫a^b sinx d x=cosa-cosb and ∫a^b cosx d x=sinb-sina for all a, b ∈ℝ.

Name: Problem 22, Chapter 7: A Course in Calculus and Real Analysis
Uploaded: 2021-11-07T01:48:14-08:00
Duration: 4 min 3 s
Channel: Steven Clarke
Description: Show that ∫a^b sinx d x=cosa-cosb and ∫a^b cosx d x=sinb-sina for all a, b ∈ℝ.

step 1 Okay, in part A I want to show this is true. To do that, I will let U equal sinex, which means D

Show that ∫a^b sinx d x=cosa-cosb and ∫a^b cosx d...

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Key Concepts

Fundamental Theorem of Calculus

This theorem connects differentiation and integration by stating that if a function is continuous over an interval, then its definite integral over that interval can be calculated using any one of its antiderivatives. Specifically, if F is an antiderivative of f, then the integral from a to b of f(x) dx equals F(b) ? F(a).

Antiderivatives of Trigonometric Functions

To evaluate the integrals of sine and cosine functions, it is essential to know their antiderivatives. The antiderivative of sin(x) is -cos(x), and the antiderivative of cos(x) is sin(x). These antiderivatives allow for the straightforward evaluation of definite integrals using the Fundamental Theorem of Calculus.

Differentiation of Trigonometric Functions

Understanding the derivative properties of sine and cosine is key to finding their antiderivatives. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These differentiation rules confirm why the antiderivative of sin(x) is -cos(x) and that of cos(x) is sin(x).

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