Integration by parts

Integration by parts is a helpful way to solve problems in calculus and mathematical analysis. It makes finding the integral of two multiplied functions simpler. Think of it as a partner version of the product rule used when finding derivatives.

The formula shows how to rewrite the integral of one function times the derivative of another. It says the integral from point a to point b of u times v prime dx can be found by calculating u times v at b, minus u times v at a, and then subtracting the integral from a to b of u prime times v dx.

This idea was first shared by mathematician Brook Taylor in 1715. There are also more complex forms of integration by parts for different kinds of integrals. A related concept called summation by parts works with sequences too.

Theorem

Integration by parts is a helpful way to solve some problems in calculus. It is used when we need to find the integral of a product of two functions.

This method comes from the product rule for differentiation. The basic idea is to change the integral of a product into a different integral that might be easier to solve. This can be useful when one function is easy to differentiate and the other is easy to integrate.

Visualization

Integration by parts can be thought of with a picture. Imagine a curve that shows how two numbers, x and y, change together. By looking at the spaces inside this curve, we can see how integration by parts helps us solve hard problems.

This way of solving works well for finding the total for an opposite function—like the logarithm or opposite angle measures—when we already know the total for the normal function. It helps because the curve’s x and y values are opposites, so we can switch between them to make the problem easier.

Applications

Integration by parts is a helpful tool in calculus. It is used to find the integral of a product of functions. This method makes hard problems easier by breaking them into smaller parts.

It works because of the product rule for differentiation. By picking the right parts of a function to change and integrate, you can often solve problems that are tricky. For instance, integrating a function like ( \frac{\ln(x)}{x^2} ) is easier with this method.

This method is also important in advanced areas of mathematics, such as harmonic analysis and operator theory. It helps show key properties and relationships in these areas.

Repeated integration by parts

Further information: Cauchy formula for repeated integration

Integration by parts can be used more than once. This helps when working with some kinds of functions. For instance, when one function is a polynomial and the other is a trigonometric function, doing integration by parts again and again can make the problem easier.

We can set up a table for this method, called "tabular integration." The table shows the derivatives of one function and the integrals of the other, and we keep going until we see a pattern. This idea was shown in the film Stand and Deliver (1988).

# i	Sign	A: derivatives u ( i ) {\displaystyle u^{(i)}}	B: integrals v ( n − i ) {\displaystyle v^{(n-i)}}
0	+	x 3 {\displaystyle x^{3}}	cos ⁡ x {\displaystyle \cos x}
1	−	3 x 2 {\displaystyle 3x^{2}}	sin ⁡ x {\displaystyle \sin x}
2	+	6 x {\displaystyle 6x}	− cos ⁡ x {\displaystyle -\cos x}
3	−	6 {\displaystyle 6}	− sin ⁡ x {\displaystyle -\sin x}
4	+	0 {\displaystyle 0}	cos ⁡ x {\displaystyle \cos x}

Higher dimensions

Integration by parts can work with functions that have many parts. It is like breaking a big puzzle into smaller pieces. This idea comes from rules about shapes and changes.

One rule helps us see how these pieces fit together. By using this rule and a way to measure areas, we can change a hard problem into easier ones. This method is useful in advanced math and physics for solving complicated equations.

Theorem

Visualization

Applications

Repeated integration by parts

Higher dimensions

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