A.J.710 said:I am just stuck on the fact that the antiderivative of cos(x) is -sin(x)
but the integral of cos(x) is a positive vale of sin(x). Can someone please explain this to me. I am two weeks behind on my calculus course now because I got stuck on integration by parts and don't understand anything after that and I am reading over my teacher's answer key and see this in one of the solutions and don't understand why. Thanks
LCKurtz said:No. That is the derivative on cos(x), not its antiderivative.
Antiderivative and integral are two concepts in calculus that are closely related. Antiderivative is the inverse operation of taking a derivative, while integral is the reverse operation of taking a derivative. In simpler terms, the antiderivative of a function is the original function, while the integral of a function is the area under the curve of that function.
The notation used for antiderivatives is f'(x), or dy/dx, while the notation used for integrals is the integral sign ∫ followed by the function or expression to be integrated, and then the variables of integration. For example, ∫f(x)dx is read as "the integral of f of x with respect to x".
The relationship between antiderivatives and integrals is that they are inverse operations of each other. This means that if a function F(x) is the antiderivative of f(x), then ∫f(x)dx = F(x) + C, where C is a constant. This is known as the Fundamental Theorem of Calculus.
To find the antiderivative of a function, you can use the basic rules of differentiation in reverse. For example, if the derivative of a function f(x) is 2x, then the antiderivative would be x2 + C. To find the integral of a function, you can use various integration techniques such as substitution, integration by parts, and partial fractions.
Antiderivatives and integrals have many practical applications in physics, engineering, and other fields. For example, they are used to calculate the area under a curve, which is useful in finding the distance traveled by an object with a changing velocity. They are also used in calculating the work done by a force, finding the volume of irregular shapes, and determining the center of mass of an object.