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Prove the formula, where $ m $ and $ n $ are positive integers.

$ \displaystyle \int_{-\pi}^{\pi} \sin mx \cos nx dx = 0 $

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If $m=n,$ then the first term in each set of brackets is zero.

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

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Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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Prove the formula, where m…

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Prove the formula, where $…

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Prove that the integral of…

we'd like to prove that the inner world from negative pika pi of sign MX Times Co. Sign an X zero where Eman enter positive imagers. So to do this, let's take a look at them two grand. Let's call that F IX. So observe that half of negative X a sign and negative eggs co sign and negative X, or which weaken right is sign negative MX Coast Negative annex and then here for the sign we can pull out the minus sign. So this's sense sign is odd. So our equivalently this means a sign of negative T. It's negative, Sci fi and co sign is even so we can ignore the minus on the inside in just right it is co sign an ex. So these air properties of signing co signed that we know from Trig and we're left with up here We're left with negative f of X. So we conclude that eh facade So also we know that anytime you have ah odd function and you integrate it from negative eight a you'LL get a zero. So in this problem we haven't no girl from negative ada oven not function So the answer will always be zero and that proves the formula

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