How to use the Integral Calculator
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Type in the integral problem to solve
To get started, type in a value of the integral problem and click «Submit» button. In a moment you will receive the calculation result. -
See a step-by-step solution
If you need to understand how the problem was solved, you can see a detailed step-by-step solution. -
Save the results of your calculation
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What is Integrals?
One of the essential tools in Calculus is Integration. We use it to find anti-derivatives, the area of two-dimensional regions, volumes, central points, among many other ways. Knowing how to use integration rules is, therefore, key to being good at Calculus.
But, let’s start with the basics; Integrals. What is Integrals?
There are two types of integrals: The indefinite integral and the definite integral. The indefinite integral f(x), which is denoted by f(x) dx, is the anti-derivative of f(x). The derivative of f(x) dx is, therefore, f(x). As you might already be aware, the derivative of a constant is always 0. For this reason, indefinite integrals are only defined up to some arbitrary constant. Consider sin(x)dx = -cos (x) + constant. The derivative of –cos(x) + constant is sin (x).
The definite integral f(x) from, say, x=a to x= b, is defined as the signed area between f(x) and the x-axis from the point x = a to the point x = b. The definite integral is denoted by a f(x) d(x).
It is important to note that both the definite and indefinite integrals are interlinked by the fundamental theorem of calculus. The theorem states that if f(x) is continuous on [a,b], and F(x) is its continuous indefinate integral, then a f(x) dx= F(b) – F(a). This, therefore, means that 0 sin(x) dx = {-cos(π)} – {-cos(0)} = 2.
Note, however, that sometimes, you may be required to find an approximation to a definite integral. The most common way to do this is to have several thin rectangles under the curve from the initial point x = a to the last point x = b. Add the signed areas (areas of the rectangles) together, and there you go! You have the desired definite integral.
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