What's the indefinite integral formula?

In summary: I was confusing the 1/pi with the 2/pi. So 1/pi and pi/2 are constants and should be separated, right?Yes, you should always pull out constants when integrating. In this case, 2 is pulled out, but pi should also be pulled out, as it is a constant.
  • #1
afcwestwarrior
457
0

Homework Statement


evaluate the integral
∫x^2 sinpi x dx


Homework Equations


∫u dv= uv - ∫v du
integration by parts formula






The Attempt at a Solution


u=x^2 dv= sin pi x dx
du = 2x v = -cos pi x dx ? the pi is giving me trouble
 
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  • #2
afcwestwarrior said:

The Attempt at a Solution


u=x^2 dv= sin pi x dx
du = 2x v = -cos pi x dx ? the pi is giving me trouble


Why?
Would using 3.14 instead of pi would help?

Sorry, I am not used to this type of Int by parts. I make
d/dx (f(x)) = .. equation and integrate

But, seems like you having trouble integrating cos(pi*x)?
 
  • #3
i figured it out already
 
  • #4
my answer is -1/pi cos pi x + 1/pi 2x sin pi x - 2(1/pi) -cos pi x +c
but in the back of the book the answer is - 1/pi cos pi x + 2/pi^2 sin pi x + 2/pi^3 cos pi x +c
 
  • #5
afcwestwarrior said:
my answer is -1/pi cos pi x + 1/pi 2x sin pi x - 2(1/pi) -cos pi x +c
but in the back of the book the answer is - 1/pi cos pi x + 2/pi^2 sin pi x + 2/pi^3 cos pi x +c

possible that you can write out the solution?

I hate to simplify and compare but this is what I got with MATLAB if you want confirm it with book:

>> int('x^2*sin(pi*x)','x')

ans =

1/pi^3*(-pi^2*x^2*cos(pi*x)+2*cos(pi*x)+2*pi*x*sin(pi*x))
 
Last edited:
  • #6
here's me work

u=x^2 dv= sin pi x
du = 2x v = 1/pi -cos pi x dx

x^2 (1/pi) (-cos pi x) -∫2x (1/pi) -cos pi x dx or - (1/pi) x^2 (cos pi x) + (1/pi) ∫2x cos pi x dx

u=2x dv= (1/pi) cos pi x
du=2 v= (1/pi) sin pi x

(1/pi) ∫2x cos pi x dx= 2x (1/pi) sin pi x - ∫ 2(1/pi) sin pi x dx
plug this equation to the other one and you get


(1/pi) x^2 (cos pi x) + (2/pi) x sin pi x + 2/pi cos pi x + c
 
  • #7
afcwestwarrior said:
(1/pi) x^2 (cos pi x) + (2/pi) x sin pi x + 2/pi cos pi x + c

You are forgetting to divide by pi when you substitute (you are multiplying by 2 but forgetting about pi as it is 2/pi?)
and second problem is you should have cos(pi*x)/pi^2 but you have cos(pi*x)/pi

Only seems to be coefficient problem, everything else looks good!
 
  • #8
afcwestwarrior said:
my answer is
-1/pi cos pi x + 1/pi 2x sin pi x - 2(1/pi) -cos pi x +c
but in the back of the book the answer is
- 1/pi cos pi x + 2/pi^2 sin pi x + 2/pi^3 cos pi x +c​

Hi afcwestwarrior! :smile:

(have a pi: π :smile:)

The only difference is that you have 1/π everywhere, but the answer has 1/π2 or 1/π3

that's because each time you integrate a function of (πx), you must divide by π …

two integrations π2, three integrations π3. :wink:
 
  • #9
Ok, so your saying that i have to divide by pi each time i integrate a function of nx.
 
  • #10
so if that pi were a 5, then i'd have to divide by 5 each time right
 
  • #11
I get it now thanks.
 

1. What is the indefinite integral formula?

The indefinite integral formula is used in calculus to find the antiderivative of a given function. It is represented by the symbol ∫ (integral sign) and is written as ∫f(x)dx. It represents the anti-derivative of the function f(x) with respect to the variable x.

2. How do you find the indefinite integral?

To find the indefinite integral, you must first determine the function f(x) and the variable x. Then, you can use the power rule, product rule, quotient rule, or chain rule to integrate the function. You can also use integration by parts or substitution methods to find the indefinite integral.

3. What is the difference between definite and indefinite integrals?

A definite integral has specified upper and lower limits, while an indefinite integral does not. This means that a definite integral gives a single numerical value, while an indefinite integral gives a function that represents all possible antiderivatives of the given function.

4. Can you show an example of finding the indefinite integral?

Yes, for example, if we have the function f(x) = 2x^2, the indefinite integral would be ∫2x^2dx = (2/3)x^3 + C, where C is the constant of integration. This means that all antiderivatives of the function 2x^2 will have the form (2/3)x^3 + C.

5. Why is the indefinite integral important in calculus?

The indefinite integral is important because it allows us to find the original function from its derivative. It also helps us solve a variety of problems in mathematics, physics, and engineering, such as finding the area under a curve, calculating work and displacement, and solving differential equations.

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