Integrate (xsinx - cosx)/x^2 with Intro Calc Techniques

In summary, the conversation discusses a problem involving an integral that cannot be solved with elementary methods. The original integral involves the sinc function and there may be a typo in the problem. The conversation also mentions using IBP and getting stuck in a loop.
  • #1
The Head
144
2
Homework Statement
Evaluate ∫((xsin(x) - cos(x))/x^2. Hint: Use Integration by Parts for sin(x)/x

Problem also attached if that's clearer
Relevant Equations
IBP: ∫u dv = uv- ∫v du
I'm pretty confused here because after getting stuck on this problem, I tossed it into an integral calculator and it said the answer was 2 Si(x) + cos(x)/x + C. In intro calc we definitely haven't learned the Si(x) function or even gotten to any of the Taylor polynomial stuff yet.

I tried IBP for sin(x)/x and got -cos(x)/x - ∫cos(x)/x^2 dx. At first this looked promising, since that is my second term. So I had in total now: -cos(x)/x - 2∫cos(x)/x^2 dx.

To do the integral portion, I integrated again (by parts) to find -2∫cos(x)/x^2 dx= 2*cos(x)/x +2∫sin(x)/x dx.

Putting it all together, the original integral = -cos(x)/x + 2*cos(x)/x +2∫sin(x)/x dx= cos(x)/x +2∫sin(x)/x dx. I seem to be getting into a loop, where the integral of cos(x)/x^2 is related to the integral of the sin(x)/x, but can't seem to simplify the terms.

Help on how to proceed?
 

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  • #3
Thanks for confirming it can't be solved-- I'll try to see what's up with the problem.
 
  • #4
scottdave said:
Perhaps there is a typo.
I'd guess the minus sign in the numerator was supposed to be a plus sign.
 
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Likes SammyS and archaic

Related to Integrate (xsinx - cosx)/x^2 with Intro Calc Techniques

1. What is integration?

Integration is a mathematical process that involves finding the area under a curve on a graph. It is the inverse operation of differentiation, which is finding the slope of a curve at a given point.

2. What is the purpose of integrating a function?

The purpose of integrating a function is to find the total value of the function over a given interval. This can be useful in various real-world applications such as calculating displacement, velocity, and acceleration in physics or finding the total profit or loss in economics.

3. How do I integrate a function using introductory calculus techniques?

To integrate a function using introductory calculus techniques, you can use the power rule, substitution, integration by parts, or partial fractions. It is important to understand the fundamental principles of calculus and practice solving various integration problems to become proficient in these techniques.

4. What is the specific technique for integrating (xsinx - cosx)/x^2?

The specific technique for integrating (xsinx - cosx)/x^2 is substitution. By substituting u = x^2, the integral can be rewritten as ∫(sinx - cosx)/u du, which can then be solved using the power rule and trigonometric identities.

5. Are there any tips for solving this type of integration problem?

One tip for solving this type of integration problem is to carefully choose the substitution variable. In this case, choosing u = x^2 eliminates the fraction and makes the integral easier to solve. Additionally, it is important to review and understand the properties of trigonometric functions to simplify the integration process.

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