Solving Indefinite Integrals: "int (1/(sqrt -x^2 -2x))dx

In summary, an indefinite integral is the inverse operation of differentiation and is represented by the symbol "∫". To solve it, you can use techniques such as substitution, integration by parts, or partial fractions. The power rule for integration states that the integral of x^n is equal to x^(n+1) / (n+1), with the exception of -1. Indefinite integrals can have multiple solutions due to the constant of integration, but definite integrals have a unique solution when limits are specified. Solving indefinite integrals is important in mathematics and science, as well as practical applications such as calculating areas, volumes, and solving differential equations.
  • #1
Atilla1982
18
0
I have:


int (1/(sqrt -x^2 -2x))dx

so I rewrite (-x^2-2x) --> 1-(x+1)^2 and swap those two.

then I say t=x+1, and substitute that in.

So now I have:

int (1/(sqrt 1-t^2)) dt

Here I get stuck, can anyone please help?
 
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  • #2
That was an excellent substitution. Review the derivative of the arcsine function.
 

1. What is an indefinite integral?

An indefinite integral is the inverse operation of differentiation, which involves finding the original function given its derivative. It is represented by the symbol "∫" and is also known as an antiderivative.

2. How do you solve indefinite integrals?

To solve an indefinite integral, you can use integration techniques such as substitution, integration by parts, or partial fractions. In this specific case, you can use the substitution method by letting u = -x^2 -2x and then finding the integral of 1/u using the power rule.

3. What is the power rule for integration?

The power rule for integration states that the integral of x^n is equal to x^(n+1) / (n+1), where n is any real number except for -1. In this case, the integral of 1/u would be equal to ln|u| + C.

4. Can indefinite integrals have multiple solutions?

No, indefinite integrals have a constant of integration (C) which can lead to multiple solutions. However, if the limits of integration are specified, then the definite integral will have a unique solution.

5. Why is it important to solve indefinite integrals?

Solving indefinite integrals is important in mathematics and science as it allows us to determine the original function from its derivative. It is also used in many practical applications such as calculating areas, volumes, and solving differential equations.

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