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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

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