<< Chapter < Page Chapter >> Page >

Use substitution to find the antiderivative of d x 25 + 4 x 2 .

1 10 tan −1 ( 2 x 5 ) + C

Got questions? Get instant answers now!

Applying the integration formulas

Find the antiderivative of 1 9 + x 2 d x .

Apply the formula with a = 3 . Then,

d x 9 + x 2 = 1 3 tan −1 ( x 3 ) + C .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the antiderivative of d x 16 + x 2 .

1 4 tan −1 ( x 4 ) + C

Got questions? Get instant answers now!

Evaluating a definite integral

Evaluate the definite integral 3 / 3 3 d x 1 + x 2 .

Use the formula for the inverse tangent. We have

3 / 3 3 d x 1 + x 2 = tan −1 x | 3 / 3 3 = [ tan −1 ( 3 ) ] [ tan −1 ( 3 3 ) ] = π 6 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate the definite integral 0 2 d x 4 + x 2 .

π 8

Got questions? Get instant answers now!

Key concepts

  • Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.
  • Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.
  • Substitution is often required to put the integrand in the correct form.

Key equations

  • Integrals That Produce Inverse Trigonometric Functions
    d u a 2 u 2 = sin −1 ( u a ) + C
    d u a 2 + u 2 = 1 a tan −1 ( u a ) + C
    d u u u 2 a 2 = 1 a sec −1 ( u a ) + C

In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

0 3 / 2 d x 1 x 2

sin −1 x | 0 3 / 2 = π 3

Got questions? Get instant answers now!

−1 / 2 1 / 2 d x 1 x 2

Got questions? Get instant answers now!

3 1 d x 1 + x 2

tan −1 x | 3 1 = π 12

Got questions? Get instant answers now!

1 2 d x | x | x 2 1

sec −1 x | 1 2 = π 4

Got questions? Get instant answers now!

1 2 / 3 d x | x | x 2 1

Got questions? Get instant answers now!

In the following exercises, find each indefinite integral, using appropriate substitutions.

d x 9 x 2

sin −1 ( x 3 ) + C

Got questions? Get instant answers now!

d x 9 + x 2

1 3 tan −1 ( x 3 ) + C

Got questions? Get instant answers now!

d x | x | x 2 9

1 3 sec −1 ( x 3 ) + C

Got questions? Get instant answers now!

d x | x | 4 x 2 16

Got questions? Get instant answers now!

Explain the relationship cos −1 t + C = d t 1 t 2 = sin −1 t + C . Is it true, in general, that cos −1 t = sin −1 t ?

cos ( π 2 θ ) = sin θ . So, sin −1 t = π 2 cos −1 t . They differ by a constant.

Got questions? Get instant answers now!

Explain the relationship sec −1 t + C = d t | t | t 2 1 = csc −1 t + C . Is it true, in general, that sec −1 t = csc −1 t ?

Got questions? Get instant answers now!

Explain what is wrong with the following integral: 1 2 d t 1 t 2 .

1 t 2 is not defined as a real number when t > 1 .

Got questions? Get instant answers now!

Explain what is wrong with the following integral: −1 1 d t | t | t 2 1 .

Got questions? Get instant answers now!

In the following exercises, solve for the antiderivative f of f with C = 0 , then use a calculator to graph f and the antiderivative over the given interval [ a , b ] . Identify a value of C such that adding C to the antiderivative recovers the definite integral F ( x ) = a x f ( t ) d t .

[T] 1 9 x 2 d x over [ −3 , 3 ]


Two graphs. The first shows the function f(x) = 1 / sqrt(9 – x^2). It is an upward opening curve symmetric about the y axis, crossing at (0, 1/3). The second shows the function F(x) = arcsin(1/3 x). It is an increasing curve going through the origin.
The antiderivative is sin −1 ( x 3 ) + C . Taking C = π 2 recovers the definite integral.

Got questions? Get instant answers now!

[T] 9 9 + x 2 d x over [ −6 , 6 ]

Got questions? Get instant answers now!

[T] cos x 4 + sin 2 x d x over [ −6 , 6 ]


Two graphs. The first shows the function f(x) = cos(x) / (4 + sin(x)^2). It is an oscillating function over [-6, 6] with turning points at roughly (-3, -2.5), (0, .25), and (3, -2.5), where (0,.25) is a local max and the others are local mins. The second shows the function F(x) = .5 * arctan(.5*sin(x)), which also oscillates over [-6,6]. It has turning points at roughly (-4.5, .25), (-1.5, -.25), (1.5, .25), and (4.5, -.25).
The antiderivative is 1 2 tan −1 ( sin x 2 ) + C . Taking C = 1 2 tan −1 ( sin ( 6 ) 2 ) recovers the definite integral.

Got questions? Get instant answers now!

[T] e x 1 + e 2 x d x over [ −6 , 6 ]

Got questions? Get instant answers now!

In the following exercises, compute the antiderivative using appropriate substitutions.

sin −1 t d t 1 t 2

1 2 ( sin −1 t ) 2 + C

Got questions? Get instant answers now!

d t sin −1 t 1 t 2

Got questions? Get instant answers now!

tan −1 ( 2 t ) 1 + 4 t 2 d t

1 4 ( tan −1 ( 2 t ) ) 2

Got questions? Get instant answers now!

t tan −1 ( t 2 ) 1 + t 4 d t

Got questions? Get instant answers now!

sec −1 ( t 2 ) | t | t 2 4 d t

1 4 ( sec −1 ( t 2 ) 2 ) + C

Got questions? Get instant answers now!

t sec −1 ( t 2 ) t 2 t 4 1 d t

Got questions? Get instant answers now!

In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [ a , b ] . Approximate a value of C , if possible, such that adding C to the antiderivative gives the same value as the definite integral F ( x ) = a x f ( t ) d t .

[T] 1 x x 2 4 d x over [ 2 , 6 ]


A graph of the function f(x) = -.5 * arctan(2 / ( sqrt(x^2 – 4) ) ) in quadrant four. It is an increasing concave down curve with a vertical asymptote at x=2.
The antiderivative is 1 2 sec −1 ( x 2 ) + C . Taking C = 0 recovers the definite integral over [ 2 , 6 ] .

Got questions? Get instant answers now!

[T] 1 ( 2 x + 2 ) x d x over [ 0 , 6 ]

Got questions? Get instant answers now!

[T] ( sin x + x cos x ) 1 + x 2 sin 2 x d x over [ −6 , 6 ]


The graph of f(x) = arctan(x sin(x)) over [-6,6]. It has five turning points at roughly (-5, -1.5), (-2,1), (0,0), (2,1), and (5,-1.5).
The general antiderivative is tan −1 ( x sin x ) + C . Taking C = tan −1 ( 6 sin ( 6 ) ) recovers the definite integral.

Got questions? Get instant answers now!

Questions & Answers

find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
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
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
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?
no
Haliru
is derivatives of any constant no is zero
Haliru
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

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