Inverse Integral: $\int \frac{x^2\cos^{-1}\left(x\sqrt{x}\right)}{(1-x^3)^2}dx$

In summary, an inverse integral is the opposite process of taking a derivative, where the antiderivative of a function is found. Its purpose is to find the original function that was differentiated to get a given function, making it useful in solving problems in physics, engineering, and other scientific fields. To solve an inverse integral, integration techniques such as substitution, integration by parts, or partial fractions are used. The formula for an inverse integral is ∫f(x)dx = F(x) + C, also known as the fundamental theorem of calculus. However, not all inverse integrals can be solved analytically and may require numerical methods or approximations.
  • #1
juantheron
247
1
$\displaystyle \int \frac{x^2\cos^{-1}\left(x\sqrt{x}\right)}{(1-x^3)^2}dx$
 
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  • #2
integrate by parts [tex]dv = \frac{x^2}{(1-x^3)^2} [/tex] and [tex]u = \cos ^{-1} (x \sqrt{x} ) [/tex]
 

FAQ: Inverse Integral: $\int \frac{x^2\cos^{-1}\left(x\sqrt{x}\right)}{(1-x^3)^2}dx$

What is an inverse integral?

An inverse integral is an integral that involves finding the antiderivative of a function. It is the opposite process of taking a derivative, where the derivative of a function is found.

What is the purpose of an inverse integral?

The purpose of an inverse integral is to find the original function that was differentiated to get a given function. It is useful in solving problems in physics, engineering, and other scientific fields.

How do you solve an inverse integral?

To solve an inverse integral, you need to use integration techniques such as substitution, integration by parts, or partial fractions. These techniques help to simplify the integral and make it easier to find the antiderivative.

What is the formula for an inverse integral?

The formula for an inverse integral is ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration. This formula is also known as the fundamental theorem of calculus.

Can an inverse integral always be solved?

No, not all inverse integrals can be solved analytically. Some integrals are too complex and do not have a known antiderivative. In these cases, numerical methods or approximations can be used to solve the integral.

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