Rainbow Child said:Split it into two intervals, i.e. [itex](-\infty,0),\,(0,\infty)[/itex] and make a change or variables in the first one [itex]x\to-x[/itex]
Rainbow Child said:Since [itex]x\in(0,\infty)\Rightarrow |x|=x[/itex]. Now combine the two integrals.
The definite integral of an exponential function is a mathematical concept used to find the area under a curve of an exponential function between two specific points on the x-axis. It represents the sum of infinitely many infinitely thin rectangles that make up the area under the curve.
To evaluate a definite integral of an exponential function, you can use the fundamental theorem of calculus, which states that the definite integral of a function f(x) from a to b is equal to the difference of the antiderivatives of f evaluated at a and b. In simpler terms, you can find the antiderivative of the exponential function, plug in the upper and lower limits of integration, and then subtract the values.
The definite integral and derivative of an exponential function are closely related. The derivative of an exponential function is itself, and the definite integral of an exponential function is the area under the curve of that function. In other words, the derivative tells us the instantaneous rate of change of the function, while the definite integral tells us the overall change in the function over a specific interval.
Yes, the definite integral of an exponential function can be negative. This occurs when the function is below the x-axis between the upper and lower limits of integration. The negative value indicates that the area under the curve is decreasing, while a positive value would indicate an increasing area.
The definite integral of an exponential function has many real-world applications, such as calculating compound interest in finance, determining population growth in biology, and finding the total displacement of an object over time in physics. It is a powerful tool for finding the total change or accumulation of a quantity over a specific interval.