<< Chapter < Page Chapter >> Page >
  • 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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the antiderivative of d x 1 16 x 2 .

1 4 sin −1 ( 4 x ) + C

Got questions? Get instant answers now!

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

Find the indefinite integral using an inverse trigonometric function and substitution for d x 9 x 2 .

sin −1 ( x 3 ) + C

Got questions? Get instant answers now!

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

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

Questions & Answers

what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

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?