<< Chapter < Page Chapter >> Page >
Integration formulas
Differentiation Formula Indefinite Integral
d d x ( k ) = 0 k d x = k x 0 d x = k x + C
d d x ( x n ) = n x n 1 x n d n = x n + 1 n + 1 + C for n 1
d d x ( ln | x | ) = 1 x 1 x d x = ln | x | + C
d d x ( e x ) = e x e x d x = e x + C
d d x ( sin x ) = cos x cos x d x = sin x + C
d d x ( cos x ) = sin x sin x d x = cos x + C
d d x ( tan x ) = sec 2 x sec 2 x d x = tan x + C
d d x ( csc x ) = csc x cot x csc x cot x d x = csc x + C
d d x ( sec x ) = sec x tan x sec x tan x d x = sec x + C
d d x ( cot x ) = csc 2 x csc 2 x d x = cot x + C
d d x ( sin −1 x ) = 1 1 x 2 1 1 x 2 = sin −1 x + C
d d x ( tan −1 x ) = 1 1 + x 2 1 1 + x 2 d x = tan −1 x + C
d d x ( sec −1 | x | ) = 1 x x 2 1 1 x x 2 1 d x = sec −1 | x | + C

From the definition of indefinite integral of f , we know

f ( x ) d x = F ( x ) + C

if and only if F is an antiderivative of f . Therefore, when claiming that

f ( x ) d x = F ( x ) + C

it is important to check whether this statement is correct by verifying that F ( x ) = f ( x ) .

Verifying an indefinite integral

Each of the following statements is of the form f ( x ) d x = F ( x ) + C . Verify that each statement is correct by showing that F ( x ) = f ( x ) .

  1. ( x + e x ) d x = x 2 2 + e x + C
  2. x e x d x = x e x e x + C
  1. Since
    d d x ( x 2 2 + e x + C ) = x + e x ,

    the statement
    ( x + e x ) d x = x 2 2 + e x + C

    is correct.
    Note that we are verifying an indefinite integral for a sum. Furthermore, x 2 2 and e x are antiderivatives of x and e x , respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.
  2. Using the product rule, we see that
    d d x ( x e x e x + C ) = e x + x e x e x = x e x .

    Therefore, the statement
    x e x d x = x e x e x + C

    is correct.
    Note that we are verifying an indefinite integral for a product. The antiderivative x e x e x is not a product of the antiderivatives. Furthermore, the product of antiderivatives, x 2 e x / 2 is not an antiderivative of x e x since
    d d x ( x 2 e x 2 ) = x e x + x 2 e x 2 x e x .

    In general, the product of antiderivatives is not an antiderivative of a product.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Verify that x cos x d x = x sin x + cos x + C .

d d x ( x sin x + cos x + C ) = sin x + x cos x sin x = x cos x

Got questions? Get instant answers now!

In [link] , we listed the indefinite integrals for many elementary functions. Let’s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum f + g . In [link] a. we showed that an antiderivative of the sum x + e x is given by the sum ( x 2 2 ) + e x —that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if F and G are antiderivatives of any functions f and g , respectively, then

d d x ( F ( x ) + G ( x ) ) = F ( x ) + G ( x ) = f ( x ) + g ( x ) .

Therefore, F ( x ) + G ( x ) is an antiderivative of f ( x ) + g ( x ) and we have

( f ( x ) + g ( x ) ) d x = F ( x ) + G ( x ) + C .

Similarly,

( f ( x ) g ( x ) ) d x = F ( x ) G ( x ) + C .

In addition, consider the task of finding an antiderivative of k f ( x ) , where k is any real number. Since

d d x ( k f ( x ) ) = k d d x F ( x ) = k F ( x )

for any real number k , we conclude that

k f ( x ) d x = k F ( x ) + C .

These properties are summarized next.

Properties of indefinite integrals

Let F and G be antiderivatives of f and g , respectively, and let k be any real number.

Sums and Differences

( f ( x ) ± g ( x ) ) d x = F ( x ) ± G ( x ) + C

Constant Multiples

k f ( x ) d x = k F ( x ) + C

From this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see [link] b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in Introduction to Integration . In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.

Questions & Answers

determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Debdoot
A conical container of radius 10 ft and height 30 ft is filled with water to a depth of 15 ft. How much work is required to pump all the water out through a hole in the top of the container if the unit weight of the water is 62.4 lb/ft^3?
Milca Reply
hi am new here I really wants to know how the solve calculus
IBRAHIM
me too. I want to know calculation involved in calculus.
Katiba
evaluate triple integral xyz dx dy dz where the domain v is bounded by the plane x+y+z=a and the co-ordinate planes
BAGAM Reply
So how can this question be solved
Eddy
i m not sure but it could be xyz/2
Leo
someone should explain with a photo shot of the working pls
Adegoke
I think we should sort it out.
Eunice
Eunice Toe you can try it if you have the idea
Adegoke
how
Eunice
a^6÷8
Muzamil
i think a^6 ÷ 8
Muzamil
Practice Key Terms 3

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask