<< Chapter < Page Chapter >> Page >

Problem-solving strategy: integrating expressions involving a 2 + x 2

  1. Check to see whether the integral can be evaluated easily by using another method. In some cases, it is more convenient to use an alternative method.
  2. Substitute x = a tan θ and d x = a sec 2 θ d θ . This substitution yields
    a 2 + x 2 = a 2 + ( a tan θ ) 2 = a 2 ( 1 + tan 2 θ ) = a 2 sec 2 θ = | a sec θ | = a sec θ . (Since π 2 < θ < π 2 and sec θ > 0 over this interval, | a sec θ | = a sec θ . )
  3. Simplify the expression.
  4. Evaluate the integral using techniques from the section on trigonometric integrals.
  5. Use the reference triangle from [link] to rewrite the result in terms of x . You may also need to use some trigonometric identities and the relationship θ = tan −1 ( x a ) . ( Note : The reference triangle is based on the assumption that x > 0 ; however, the trigonometric ratios produced from the reference triangle are the same as the ratios for which x 0. )
This figure is a right triangle. It has an angle labeled theta. This angle is opposite the vertical side. The hypotenuse is labeled the square root of (a^2+x^2), the vertical leg is labeled x, and the horizontal leg is labeled a. To the left of the triangle is the equation tan(theta) = x/a.
A reference triangle can be constructed to express the trigonometric functions evaluated at θ in terms of x .

Integrating an expression involving a 2 + x 2

Evaluate d x 1 + x 2 and check the solution by differentiating.

Begin with the substitution x = tan θ and d x = sec 2 θ d θ . Since tan θ = x , draw the reference triangle in the following figure.

This figure is a right triangle. It has an angle labeled theta. This angle is opposite the vertical side. The hypotenuse is labeled the square root of (1+x^2), the vertical leg is labeled x, and the horizontal leg is labeled 1. To the left of the triangle is the equation tan(theta) = x/1.
The reference triangle for [link] .

Thus,

d x 1 + x 2 = sec 2 θ sec θ d θ Substitute x = tan θ and d x = sec 2 θ d θ . This substitution makes 1 + x 2 = sec θ . Simplify. = sec θ d θ Evaluate the integral. = ln | sec θ + tan θ | + C Use the reference triangle to express the result in terms of x . = ln | 1 + x 2 + x | + C .

To check the solution, differentiate:

d d x ( ln | 1 + x 2 + x | ) = 1 1 + x 2 + x · ( x 1 + x 2 + 1 ) = 1 1 + x 2 + x · x + 1 + x 2 1 + x 2 = 1 1 + x 2 .

Since 1 + x 2 + x > 0 for all values of x , we could rewrite ln | 1 + x 2 + x | + C = ln ( 1 + x 2 + x ) + C , if desired.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluating d x 1 + x 2 Using a different substitution

Use the substitution x = sinh θ to evaluate d x 1 + x 2 .

Because sinh θ has a range of all real numbers, and 1 + sinh 2 θ = cosh 2 θ , we may also use the substitution x = sinh θ to evaluate this integral. In this case, d x = cosh θ d θ . Consequently,

d x 1 + x 2 = cosh θ 1 + sinh 2 θ d θ Substitute x = sinh θ and d x = cosh θ d θ . Substitute 1 + sinh 2 θ = cosh 2 θ . = cosh θ cosh 2 θ d θ cosh 2 θ = | cosh θ | = cosh θ | cosh θ | d θ | cosh θ | = cosh θ since cosh θ > 0 for all θ . = cosh θ cosh θ d θ Simplify. = 1 d θ Evaluate the integral. = θ + C Since x = sinh θ , we know θ = sinh −1 x . = sinh −1 x + C .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding an arc length

Find the length of the curve y = x 2 over the interval [ 0 , 1 2 ] .

Because d y d x = 2 x , the arc length is given by

0 1 / 2 1 + ( 2 x ) 2 d x = 0 1 / 2 1 + 4 x 2 d x .

To evaluate this integral, use the substitution x = 1 2 tan θ and d x = 1 2 sec 2 θ d θ . We also need to change the limits of integration. If x = 0 , then θ = 0 and if x = 1 2 , then θ = π 4 . Thus,

0 1 / 2 1 + 4 x 2 d x = 0 π / 4 1 + tan 2 θ 1 2 sec 2 θ d θ After substitution, 1 + 4 x 2 = tan θ . Substitute 1 + tan 2 θ = sec 2 θ and simplify. = 1 2 0 π / 4 sec 3 θ d θ We derived this integral in the previous section. = 1 2 ( 1 2 sec θ tan θ + ln | sec θ + tan θ | ) | 0 π / 4 Evaluate and simplify. = 1 4 ( 2 + ln ( 2 + 1 ) ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Rewrite x 3 x 2 + 4 d x by using a substitution involving tan θ .

32 tan 3 θ sec 3 θ d θ

Got questions? Get instant answers now!

Integrating expressions involving x 2 a 2

The domain of the expression x 2 a 2 is ( , a ] [ a , + ) . Thus, either x < a or x > a . Hence, x a 1 or x a 1 . Since these intervals correspond to the range of sec θ on the set [ 0 , π 2 ) ( π 2 , π ] , it makes sense to use the substitution sec θ = x a or, equivalently, x = a sec θ , where 0 θ < π 2 or π 2 < θ π . The corresponding substitution for d x is d x = a sec θ tan θ d θ . The procedure for using this substitution is outlined in the following problem-solving strategy.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask