In this article we will understand the Integral of E^x, Integral Calculator Step by Step (Read more).The Integral function is nothing but the inverse of the differentiation and these both play an important role in calculus.
Integral of E^x and Integral Calculator Step by Step
We will see how the integral is evolved with a simple definition.
Let E be a subset of R. let f: E→ R be a function, if there is a function F on E such that F'(x) = f(x) for all x belongs to E, then we call F an anti derivative or integral of f or a primitive of f.
Let us see an example:
Derivative of the trigonometric function (sin x) = cos x, where x belongs to R. hence if f is the function given by f(x) = cos x, x belongs to R, then the function F given by F(x) =sinx, x belongs to R is an integral of f on E, then for any real number k,
we have (F + K)'(x) = f(x) for all x belongs to E
Here, F + K is also an integral of f, thus in the above example, if c is any real constant then the function G is given by
G(x) = sin x +c, x belongs to R is also an integral of cos x.
The standard forms and properties of integrals are :
1. We know that derivative of (xn+1/n+1) = xn for n not equal to -1. Hence, if n not equal to -1, we have ∫xn dx = (xn+1/n+1) + c , where c is a constant.
2. We know that derivative of (log x) = 1/x if x > 0. Hence, ∫1/x dx = log ǀxǀ +c where c is a constant .
Some of the properties of integrals are:
1. since the functions f and g are have integrals on I ,then f + g has an integral on I and then
∫(f + g)(x) dx = ∫f(x) dx + ∫g(x) dx +c.
∫(kf)(x) dx = k∫f(x) dx +c
Integral of E^x
1. As we know that the differentiation of ex is equal to ex.
2. Now apply the integral on the d/dx ( ex)
3. As we all know that the differentiation and integration are inverse to each other so both will cancel .
4. The remaining ex + c is the answer .where c is the constant of integration for all indefinite integrals .
Therefore integral of ex is equal to ex +c. Here is an example problem based on the integral of e x:
Solve integral of 5ex dx?
Integral Calculator Step by Step
We know that differentiation of ex is ex so
d/dx 5 ex = 5 ex. Now applying integration on ∫5 ex dx
As we know that integration and differentiation are inverse and they both cancel, giving us the result as
∫5ex dx = 5∫ex dx = 5ex +c where c is the constant of integration.