Indefinite Integrals - which method is preferred?

In summary: If they expanded the final polynomial, which they didn't need to do, then it was probably just to make it clear that the results are the same as the way you did it, just with a modified constant.
  • #1
1MileCrash
1,342
41

Homework Statement



[itex]\int (x+1)^2 dx[/itex]

Homework Equations





The Attempt at a Solution



I am just getting into this, and this is a simple problem, but my book and I took two separate routes. My question, essentially, is does any constant you get just "combine" with the "any constant" C?

I went with:

[itex]\int (x+1)^2 dx = \int x^2+2x+1 dx[/itex]

Which yields
[itex]\frac{1}{3}x^3+x^2+x+C[/itex]

Now, my textbook took a wildly different method, with u substitution and arrived at:
[itex]\frac{(x+1)^3}{3}+C[/itex]

These equations are not identical, but their derivatives are. So, they are both solutions and they are both the same. My gripe with the textbook solution is that, their answer can be brought to mine but with another constant. Why would they do that?

[itex]\frac{(x+1)^3}{3}+C[/itex]

[itex]=\frac{x^3+x^2+3x+1}{3}+C[/itex]

[itex]\frac{1}{3}x^3+x^2+x + \frac{1}{3} + C[/itex]

Why would they give that as a final result? Their answer is just mine with another constant hidden in. Is there are reason for them to do that?
 
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  • #2
Their method can easily solve the integral of (x+1)^100 as (x+1)^101/101+C. There's really no profit in expanding the polynomial. Yours would do it too, but it would be painful. That's why they are doing it that way.
 
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  • #3
Dick said:
Their method can easily solve the integral of (x+1)^100. Yours would do it too, but it would be painful. That's why they are doing it that way.

I would use their method in that case, though.

Or are you suggesting that they just did it that way to show the process?
 
  • #4
1MileCrash said:
I would use their method in that case, though.

Or are you suggesting that they just did it that way to show the process?

If they expanded the final polynomial, which they didn't need to do, then it was probably just to make it clear that the results are the same as the way you did it, just with a modified constant. If you just write the answer as (x+1)^3/3+C, that's already easier than expanding and integrating term by term, isn't it? In general, I prefer to do things the easiest way, even though I know harder ways will work as well.
 
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Related to Indefinite Integrals - which method is preferred?

What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the antiderivative of a function. It is denoted by the symbol ∫, and is used to find the original function from its derivative.

What is the preferred method for solving indefinite integrals?

The preferred method for solving indefinite integrals is by using the method of substitution. This involves substituting a variable in the integral with another expression, making it easier to evaluate.

When should the method of integration by parts be used?

The method of integration by parts should be used when the integrand is a product of two functions. This method involves using the product rule for derivatives in reverse.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not. Definite integrals are used to find the area under a curve, while indefinite integrals are used to find the original function from its derivative.

How can I check my answer when solving an indefinite integral?

You can check your answer by taking the derivative of the integral you have solved. If the derivative matches the original function, then your answer is correct. You can also use online calculators or graphing software to visualize the integral and its derivative.

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