Can An Integral Have 2 Answers?

How do you explain an integral?

In calculus, an integral is the space under a graph of an equation (sometimes said as “the area under a curve”).

An integral is the reverse of a derivative, and integral calculus is the opposite of differential calculus.

A derivative is the steepness (or “slope”), as the rate of change, of a curve..

What is the integral of COSX?

Integrals of trig functions can be found exactly as the reverse of derivatives of trig functions. The integral of sinx is −cosx+C and the integral of cosx is sinx+C.

What exactly is an integral?

The term “integral” can refer to a number of different concepts in mathematics. … In calculus, an integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus.

What does a definite integral tell you?

The definite integral gives you a SIGNED area, meaning that areas above the x-axis are positive and areas below the x-axis are negative. That is why if you integrate y=sin(x) from 0 to 2Pi, the answer is 0.

Why derivative of x2 is 2x?

So what does ddxx2 = 2x mean? It means that, for the function x2, the slope or “rate of change” at any point is 2x. So when x=2 the slope is 2x = 4, as shown here: Or when x=5 the slope is 2x = 10, and so on.

What does a double integral give you?

Double integrals are a way to integrate over a two-dimensional area. Among other things, they lets us compute the volume under a surface.

Why do we use triple integrals?

Triple integrals are the analog of double integrals for three dimensions. They are a tool for adding up infinitely many infinitesimal quantities associated with points in a three-dimensional region.

How do you know when to use integration by parts?

Integration by parts is for functions that can be written as the product of another function and a third function’s derivative. A good rule of thumb to follow would be to try u-substitution first, and then if you cannot reformulate your function into the correct form, try integration by parts.

What is the derivative of 2x?

To find the derivative of 2x, we can use a well-known formula to make it a very simple process. The formula for the derivative of cx, where c is a constant, is given in the following image. Since the derivative of cx is c, it follows that the derivative of 2x is 2.

Why is integration so hard?

Integration is generally much harder than differentiation. This little demo allows you to enter a function and then ask for the derivative or integral. You can also generate random functions of varying complexity. … If integration seems hard – that’s because it really is!

What is the integration of 1?

The definite integral of 1 is the area of a rectangle between x_lo and x_hi where x_hi > x_lo. In general, the indefinite integral of 1 is not defined, except to an uncertainty of an additive real constant, C. However, in the special case when x_lo = 0, the indefinite integral of 1 is equal to x_hi.

What is integral 2x?

For example, what is the integral of 2x? You already know the derivative of x2 is 2x, so the integral of 2x is x2.

Can a function have multiple integrals?

Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on).

Can you split a definite integral?

Internal addition In other words, you can split a definite integral up into two integrals with the same integrand but different limits, as long as the pattern shown in the rule holds.

What is the integration rule?

This rule is essentially the inverse of the power rule used in differentiation, and gives us the indefinite integral of a variable raised to some power. Just to refresh your memory, the integration power rule formula is as follows: ∫ ax n dx = a. x n+1.

What is the derivative of sin2x?

In a similar way, the derivative of sin(2x) with respect to 2x is cos(2x)….Using the chain rule to find the derivative of sin(2x)sin2x► Derivative of sin2x = 2cos(2x)sin (2x)► Derivative of sin (2x) = 2cos(2x)2 more rows•Oct 6, 2020

What is the integral sign called?

Historical notation The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for “sum” or “total”).