Methods for doing integral, Mathematics

Assignment Help:

There are really three various methods for doing such integral.

Method 1:

This method uses a trig formula as,

 ∫sin(x) cos(x) dx = ½ ∫sin(2x) dx = -(1/4) cos(2x) + c

Method 2:

This method uses the substitution as,

u = cos(x)                                                         du = - sin(x)dx

∫sin(x) cos(x) dx = -∫ u du = -½ u2 + c2 = -(1/2) cos2(x) + c2

Method 3:

Now there is another substitution which could be done here as,

u = sin (x)                                                        du = cos (x)dx

∫sin(x) cos(x) dx = ∫ u du = ½ u2 + c3 = (1/2) sin2(x) + c3

Therefore, we've found three various answer each with a different constant of integration.  Though, as per the fact above these three answers must only be different by a constant because they all have similar derivative.

Actually they do only be different by a constant. We will require the following trig formulas to prove that.

cos (2x) = cos2(x) - sin2(x)                               cos2(x) + sin2(x) = 1

Start with the solution from the first method and utilize the double angle formula as above.

-(1/4) (cos2(x) - sin2(x)) + c1

Here, from the second identity above we contain,

-(1/4) (cos2(x) - (1 - cos2(x))) + c1 = -(1/4) (2cos2(x) - 1) + c1

= -(1/2) cos2(x) + (¼) + c1

It is then answer we found from the second method along with a slightly differ constant. Though,

c2 = ¼ + c1

We can do a same manipulation to find the answer from the third method as given. Again, starting with the solution from the first method utilize the double angle formula and after that substitute in for the cosine in place of the sine using,

cos2(x) = 1 - sin2(x)

Doing this provides,

-(1/4)( 1 - sin2(x)) - sin2(x) + c1 = -(1/4)(1 - 2 sin2(x)) + c1

 = (1/2) sin2(x) - (¼) + c1

it is the answer from the third method along with a different constant and again we can associate the two constants with,

c3 =- (¼) + c1

Therefore, what have we learned here? Hopefully we have seen that constants of integration are significant and we cannot forget about them. We frequently don't work with them in a Calculus I course, until now without a good understanding of them we would be hard pressed to know how integration methods differ and apparently make different answers.


Related Discussions:- Methods for doing integral

Describe the laws of sines, Q. Describe the Laws of Sines? Ans. Up...

Q. Describe the Laws of Sines? Ans. Up to now we have dealt exclusively with right triangles.  The Law of Sines and the Law of Cosines are used to solve  oblique triangles

Intersection of perpendicular tangents of hyperbola., If angle between asym...

If angle between asymtotes of hyperbola x^2/a^2-y^2/b^=1 is 120 degrees and product of perpendicular drawn from foci upon its any tangent is 9. Then find the locus of point of inte

Math homework help, I need help witth my homework can you help please

I need help witth my homework can you help please

Circls, in a given figure a,b,c and d are points on a circle such that ABC ...

in a given figure a,b,c and d are points on a circle such that ABC =40 and DAB= 60 find the measure of DBA

Ratio test - sequences and series, Ratio Test In this part we are goin...

Ratio Test In this part we are going to take a look at a test that we can make use to see if a series is absolutely convergent or not.  Remind that if a series is absolutely c

Using pythagorean theorem solve z 2 = ( x + y )2 + 3502, Two people on bik...

Two people on bikes are at a distance of  350 meters.  Person A begin riding north at a rate of 5 m/sec and 7 minutes later on Person B begin riding south at 3 m/sec.  Determine th

Earth Day Bags, #question.I headed into Target in Webster, NY for an advert...

#question.I headed into Target in Webster, NY for an advertized free Earth Day Bag in (local newspaper and on your entrance store doors) and at 10:30 a.m. on Sunday, April 22nd, th

Mechanics, find the composition of the simple harmonic motion of the same p...

find the composition of the simple harmonic motion of the same period in the perpendicular directions

Congruences, Suppose m be a positive integer, then the two integer a and b ...

Suppose m be a positive integer, then the two integer a and b called congurent modulo m ' if a - b is divisible by m i.e.  a - b = m where is an positive integer. The congru

#algebra, what is the answer of 6_5x9_4x3(1_2)

what is the answer of 6_5x9_4x3(1_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