Example of integrals involving trig functions, Mathematics

Assignment Help:

Example of Integrals Involving Trig Functions

Example: Estimate the following integral.

∫ sin5 x dx

Solution

This integral no longer contains the cosine in it that would permit us to make use of the substitution that we used above.  Hence, that substitution won't work and we are going to have to find out other way of doing this kind of integral.

 

Let us first notice that we could write the integral like this,

∫ sin5 x dx = ∫ sin4 x sin x dx = ∫ (sin2 x)2 sin x dx

Here we recall the trig identity,

cos2 x + sin2 x = 1

⇒         sin2 x = 1 - cos2 x

Along with this identity the integral can be written as,

∫ sin5 x dx = ∫ (1- cos2x)2 sin x dx

and we can now make use of the substitution u = cos x .  Doing this gives us,

∫ sin5 x dx = -∫ (1 - u2)2 du

= -∫1 - 2u2 + u4 du

= - (u - 2 u3 +1 u5) + c

= - cos x + 2 cos3 x - 1/5 cos5 x + c

Thus, with a little rewriting on the integrand we were capable to reduce this to a fairly simple substitution.


Related Discussions:- Example of integrals involving trig functions

Lognormal distribution, The Lognormal Distribution If ln(X) is a normal...

The Lognormal Distribution If ln(X) is a normally distributed random variable, then X is said to be a lognormal variable. If P1, P2, P3, ... are the prices of a scrip in per

Explain combining negative signs in integers, Explain Combining Negative Si...

Explain Combining Negative Signs in integers? You've learned about positive and negative integers. BASICS :   When you place a negative sign in front of an integer, you get

Unit circle, Unit circle A circle centered at the origin with radius 1 ...

Unit circle A circle centered at the origin with radius 1 (i.e. this circle) is called as unit circle.  The unit circle is very useful in Trigonometry. (b) x 2 + ( y - 3) 2

Adding integers, Do you subtract when you add integers.

Do you subtract when you add integers.

Determine principal strains and direction , A 100 by 150 mm rectangular pla...

A 100 by 150 mm rectangular plate is deformed as shown in the following figure. All dimensions shown in the figure are in millimeters.  Determine at point Q: (a) the strain compone

Sketch the exponental graph of f( x )=2x and g( x )= 1/2 , Example Sketc...

Example Sketch the graph of following f( x ) = 2x  and  g( x ) = ( 1 /2) x Solution Let's firstly make a table of values for these two functions. Following is

How far did the ?rst arrow goes, From a fixed point directly in front of th...

From a fixed point directly in front of the center of a bull's eye, Kim aims two arrows at the bull's eye. The first arrow nicks one point on the edge of the bull's eye; the second

Theory of indices, In algebra knowing that 2 3 = 8 is not sufficient...

In algebra knowing that 2 3 = 8 is not sufficient. Equally important to know is what would be the result if quantities like 2 3 . 2 -4 . 2 6 or  3 7 / 3 2

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd