16
Oc
Integration Examples
16.10.2013 16:22

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?

Solution:

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.

Rational Numbers
Permutations

Comments


Datenschutzerklärung
Make your free website at Beep.com
 
The responsible person for the content of this web site is solely
the webmaster of this website, approachable via this form!