Question #5791

Expert sir please explain integration by parts rules.

Expert's answer

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate it into a product of two functions ƒ(x)g(x) such that the integral produced by the integration by parts formula is easier to evaluate than the original one. The following form is useful in illustrating the best strategy to take:

int(f*g)dx = f*int(g)dx - int(f'*int(g)dx)dx.

Note that on the right-hand side, ƒ is differentiated and g is integrated; consequently it is useful to choose ƒ as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated. As a simple example, consider:

int(ln(x)/x²)dx.

Since the derivative of ln x is 1/x, we make this part of ƒ; since the anti-derivative of 1/x2 is −1/x, we make this part of g. The formula now yields:

int(ln(x)/x²)dx = -ln(x)/x - int((1/x)(-1/x))dx.

The remaining integral of −1/x2 can be completed with the power rule and is 1/x.

int(f*g)dx = f*int(g)dx - int(f'*int(g)dx)dx.

Note that on the right-hand side, ƒ is differentiated and g is integrated; consequently it is useful to choose ƒ as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated. As a simple example, consider:

int(ln(x)/x²)dx.

Since the derivative of ln x is 1/x, we make this part of ƒ; since the anti-derivative of 1/x2 is −1/x, we make this part of g. The formula now yields:

int(ln(x)/x²)dx = -ln(x)/x - int((1/x)(-1/x))dx.

The remaining integral of −1/x2 can be completed with the power rule and is 1/x.

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