Integration by parts [ x^3 * sqrt (1 - x^2)

In summary, the individual has attempted various methods to solve a difficult integral, including using a derivative value and trying substitution with the pythagorean identity. However, these methods did not yield a successful result. Ultimately, they were able to solve the integral using the method of integration by parts with u=x^2 and dv=x\sqrt{1-x^2}dx. They express gratitude for the assistance and prefer to understand the process rather than simply obtaining the answer.
  • #1
amcloughlan
2
0
I have tried pretty much every method I can think of to solve this integral but I haven't managed to get much luck. I used a derivative value (U) of x^3 and managed to get a x^5 term inside the next part and there is no easy way to get a derivative for the square root of 1-x^2.

I tried subbing in x=sinx but that didn't work either after using the pythagorean identity to get cosx after removing the square root. I worked it out got jibberish as an answer.

I'd rather know how I could go about doing this rather than get an answer! Thanks. :)
 
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  • #2
Try by parts with [itex]u=x^2[/itex] and [itex]dv=x\sqrt{1-x^2}dx[/itex]...and post your work if you get stuck
 
  • #3
got it! Thank you very much.
 

FAQ: Integration by parts [ x^3 * sqrt (1 - x^2)

What is Integration by parts?

Integration by parts is a technique used in calculus to find the integral of a product of two functions. It involves breaking down the integral into simpler parts and using the product rule of differentiation in reverse to solve for the integral.

How do you use Integration by parts to solve integrals?

To use integration by parts, you first identify the two functions in the integral and label one as u and the other as dv. Then, you use the product rule to find the derivative of u and the antiderivative of dv. Finally, you plug these values into the integration by parts formula to solve for the integral.

What is the formula for Integration by parts?

The integration by parts formula is ∫u dv = uv - ∫v du, where u and v are the two functions in the integral and du and dv are their derivatives and antiderivatives, respectively.

When should I use Integration by parts to solve an integral?

Integration by parts is most useful when the integral involves a product of functions, one of which can be easily integrated while the other can be easily differentiated. It is also helpful when the integral involves a polynomial multiplied by a trigonometric function or an exponential function.

Can Integration by parts be used for any type of integral?

No, integration by parts is not always applicable to every integral. It is most effective when the integral involves a product of functions, but there are other techniques that may be more useful for different types of integrals, such as substitution or partial fractions.

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