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

Brian 100-yard dash time was 2.68 what is the school record, Brian's 100-ya...

Brian's 100-yard dash time was 2.68 seconds more than one school record. Brian's time was 13.4 seconds. What is the school record? The school record is less than Brian's time.

Give an equations with the variable on both sides, Give an Equations with t...

Give an Equations with the variable on both sides ? Many equations that you encounter will have variables on both sides. Some of these equations will even contain grouping sy

Find the sum of given equation upto n limit, Find the sum of (1 - 1/n ) + (...

Find the sum of (1 - 1/n ) + (1 - 2/n ) + (1 - 3/n ) ....... upto n terms. Ans: (1 - 1/n ) + (1 - 2/n ) - upto n terms   ⇒[1+1+.......+n terms] - [ 1/n + 2/n +....+

What is the total balance of an account after 18 months, A certain bank pay...

A certain bank pays 3.4% interest per year for a certificate of deposit, or CD. What is the total balance of an account after 18 months along with an initial deposit of $1,250?

What is stem-and-leaf plots, Q. What is Stem-and-Leaf Plots? Ans. ...

Q. What is Stem-and-Leaf Plots? Ans. A stem-and-leaf plot is a table that provides a quick way to arrange a set of data and view its shape, or distribution. Each data val

Solve and graph equation of parabola, SOLVE AND GRAPH THE PARABOLA NOTE: W...

SOLVE AND GRAPH THE PARABOLA NOTE: WRITE YOUR SOLUTIONS AND COMPLETE EQUATION OF GRAPH SPOINTS EACH 1. V(0,0) (0.2) P-2 2. V(0,0) E-5,0) P=-5 3. V(4-3) F(4,-2) P=1 4. V-1,5)

What was joe's approximate raw act score, Using the same mean and standard ...

Using the same mean and standard deviation from problem 10 (mean m = 20.1 and a standard deviation s = 5.8). Joe was informed that he scored at the 68 th percentile on the ACT, wh

One-to-one correspondence in learning maths, How does your answer to this q...

How does your answer to this question compare with mine, which follows? i) To begin with, 1 laid the beads out in a row for counting, so that I wouldn't leave any out or count a

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