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If $ \displaystyle \int_{-\infty}^\infty f(x)\ dx $ is convergent and $ a $ and $ b $ are real numbers, show that$$ \int_{-\infty}^a f(x)\ dx + \int_a^\infty f(x)\ dx = \int_{-\infty}^b f(x)\ dx + \int_b^\infty f(x)\ dx $$

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

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.

02:36

If $\int_{-\infty}^{\infty…

02:08

0:00

If $\int_{-5}^{\infty} f(x…

05:51

$$\begin{array}{l}{\te…

05:47

Rewriting an Integral Let …

04:09

Show that if $f(x)$ is int…

03:03

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