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  • Integrate functions resulting in inverse trigonometric functions

In this section we focus on integrals that result in inverse trigonometric functions. We have worked with these functions before. Recall from Functions and Graphs that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in Derivatives , we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.

Integrals that result in inverse sine functions

Let us begin this last section of the chapter with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.

Rule: integration formulas resulting in inverse trigonometric functions

The following integration formulas yield inverse trigonometric functions:

  1. d u a 2 u 2 = sin −1 u a + C

  2. d u a 2 + u 2 = 1 a tan −1 u a + C

  3. d u u u 2 a 2 = 1 a sec −1 u a + C


Let y = sin −1 x a . Then a sin y = x . Now let’s use implicit differentiation. We obtain

d d x ( a sin y ) = d d x ( x ) a cos y d y d x = 1 d y d x = 1 a cos y .

For π 2 y π 2 , cos y 0 . Thus, applying the Pythagorean identity sin 2 y + cos 2 y = 1 , we have cos y = 1 = sin 2 y . This gives

1 a cos y = 1 a 1 sin 2 y = 1 a 2 a 2 sin 2 y = 1 a 2 x 2 .

Then for a x a , we have

1 a 2 u 2 d u = sin −1 ( u a ) + C .

Evaluating a definite integral using inverse trigonometric functions

Evaluate the definite integral 0 1 d x 1 x 2 .

We can go directly to the formula for the antiderivative in the rule on integration formulas resulting in inverse trigonometric functions, and then evaluate the definite integral. We have

0 1 d x 1 x 2 = sin −1 x | 0 1 = sin −1 1 sin −1 0 = π 2 0 = π 2 .
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Find the antiderivative of d x 1 16 x 2 .

1 4 sin −1 ( 4 x ) + C

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Finding an antiderivative involving an inverse trigonometric function

Evaluate the integral d x 4 9 x 2 .

Substitute u = 3 x . Then d u = 3 d x and we have

d x 4 9 x 2 = 1 3 d u 4 u 2 .

Applying the formula with a = 2 , we obtain

d x 4 9 x 2 = 1 3 d u 4 u 2 = 1 3 sin −1 ( u 2 ) + C = 1 3 sin −1 ( 3 x 2 ) + C .
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Find the indefinite integral using an inverse trigonometric function and substitution for d x 9 x 2 .

sin −1 ( x 3 ) + C

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Evaluating a definite integral

Evaluate the definite integral 0 3 / 2 d u 1 u 2 .

The format of the problem matches the inverse sine formula. Thus,

0 3 / 2 d u 1 u 2 = sin −1 u | 0 3 / 2 = [ sin −1 ( 3 2 ) ] [ sin −1 ( 0 ) ] = π 3 .
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Integrals resulting in other inverse trigonometric functions

There are six inverse trigonometric functions. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. The only difference is whether the integrand is positive or negative. Rather than memorizing three more formulas, if the integrand is negative, simply factor out −1 and evaluate the integral using one of the formulas already provided. To close this section, we examine one more formula: the integral resulting in the inverse tangent function.

Finding an antiderivative involving the inverse tangent function

Find an antiderivative of 1 1 + 4 x 2 d x .

Comparing this problem with the formulas stated in the rule on integration formulas resulting in inverse trigonometric functions, the integrand looks similar to the formula for tan −1 u + C . So we use substitution, letting u = 2 x , then d u = 2 d x and 1 / 2 d u = d x . Then, we have

1 2 1 1 + u 2 d u = 1 2 tan −1 u + C = 1 2 tan −1 ( 2 x ) + C .
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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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?

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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