Why is the Fundamental Theorem of Calculus Part 1 and 2 useful for computing integrals?

The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See Note. The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula.

What is antiderivative sin?

The general antiderivative of sin(x) is −cos(x)+C . With an integral sign, this is written: ∫sin(x) dx=−cos(x)+C .

Why don’t we use integration constant in definite integration?

5 Answers. For any C, f(x)+C is an antiderivative of f′(x). These are two different things, so there is no reason to include C in a definite integral.

Why do we add constant of integration C to our integral?

In order to include all antiderivatives of f(x) , the constant of integration C is used for indefinite integrals. The importance of C is that it allows us to express the general form of antiderivatives.

Why the fundamental theorem of calculus makes sense?

There is a reason it is called the Fundamental Theorem of Calculus. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative.

What happens if you take the derivative of an integral?

Essentially, we’re just taking the derivative of an integral. In other words, the derivative of an integral of a function is just the function. Basically, the two cancel each other out like addition and subtraction.

What is the derivative of an integral with bounds?

Restating the Fundamental Theorem The following is a restatement of the Fundamental Theorem. That is, the derivative of a definite integral of f whose upper limit is the variable x and whose lower limit is the constant a equals the function f evaluated at x. This is true regardless of the value of the lower limit a.

How do you evaluate integrals without using calculus?

Use the technique of completing the square to evaluate the following integrals. Evaluate the integral without using calculus: Evaluate the integral using two different substitutions. First, let and evaluate using trigonometric substitution. Second, let and use trigonometric substitution.

How does the fundamental theorem of calculus justify the procedure?

The fundamental theorem of calculus justifies the procedure by computing the difference between the antiderivative at the upper and lower limits of the integration process. In this article, let us discuss the first, and the second fundamental theorem of calculus, and evaluating the definite integral using the theorems in detail.

What is the first fundamental theorem of integral calculus?

First fundamental theorem of integral calculus states that “Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function. Then A′ (x) = f (x), for all x ∈ [a, b]”.

What happens to x x in a differential equation when you substitute?

Well, upon computing the differential and actually performing the substitution every x x in the integral (including the x x in the dx d x) must disappear in the substitution process and the only letters left should be u u ’s (including a du d u) and we should be left with an integral that we can do.