How can we evaluate this indefinite integral of a definite integral?

In summary, the given indefinite integral is equal to the absolute value of the difference between the square root of x and the square root of 1-x, and can be solved by splitting the integral into two parts and using the appropriate substitution for each part.
  • #1
juantheron
247
1
Evaluation of Indefinite Integral $\displaystyle \int_{0}^{1} \sqrt{1-2\sqrt{x-x^2}}dx$

$\bf{My\; Try::}$ We can write the given Integral as

$\displaystyle \int_{0}^{1}\sqrt{\left(\sqrt{x}\right)^2+\left(\sqrt{1-x}\right)^2-2\sqrt{x}\cdot \sqrt{1-x}}dx$

So Integral Convert into $\displaystyle \int_{0}^{1}\left|\sqrt{x}-\sqrt{1-x}\right|dx$

Now How can i solve after that , explanation Required.

Thanks
 
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  • #2
jacks said:
Evaluation of Indefinite Integral $\displaystyle \int_{0}^{1} \sqrt{1-2\sqrt{x-x^2}}dx$

$\bf{My\; Try::}$ We can write the given Integral as

$\displaystyle \int_{0}^{1}\sqrt{\left(\sqrt{x}\right)^2+\left(\sqrt{1-x}\right)^2-2\sqrt{x}\cdot \sqrt{1-x}}dx$

So Integral Convert into $\displaystyle \int_{0}^{1}\left|\sqrt{x}-\sqrt{1-x}\right|dx$

Now How can i solve after that , explanation Required.

Thanks

... the successive step is easy...

$\displaystyle \int_{0}^{1} |\sqrt{x} - \sqrt{1 - x}|\ dx = \int_{0}^{\frac{1}{2}} (\sqrt{1 - x} - \sqrt{x})\ dx + \int_{\frac{1}{2}}^{1} (\sqrt{x} - \sqrt{1 - x})\ d x $

Kind regards

$\chi$ $\sigma$
 
  • #3
This is actually a DEFINITE integral...
 

FAQ: How can we evaluate this indefinite integral of a definite integral?

What is an indefinite integral?

An indefinite integral is a mathematical concept used in calculus to find the antiderivative of a function. It represents the family of all possible antiderivatives of a given function and is denoted by the integral sign without any upper and lower limits.

How is an indefinite integral different from a definite integral?

A definite integral has specific upper and lower limits, while an indefinite integral does not. This means that the value of a definite integral represents the area under the curve between two specific points, while the value of an indefinite integral represents the family of all possible antiderivatives of a function.

What is the process of finding an indefinite integral?

The process of finding an indefinite integral involves using integration techniques such as substitution, integration by parts, or partial fractions to find the antiderivative of a given function. This antiderivative is then represented by the indefinite integral symbol ∫ f(x) dx.

Can any function have an indefinite integral?

No, not all functions have an indefinite integral. For a function to have an indefinite integral, it must be continuous on its domain. Additionally, some functions may have an indefinite integral that cannot be expressed in terms of elementary functions, such as trigonometric or logarithmic functions.

What is the relationship between the derivative and the indefinite integral?

The derivative and indefinite integral are inverse operations of each other. This means that if a function has an indefinite integral, its derivative will be the original function. This is known as the Fundamental Theorem of Calculus and is a fundamental concept in calculus.

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