Integration by Parts

INTRODUCTION TO A SPECIAL METHOD OF INTEGRATION

Integration by parts (Latin: per partes) is one of the more important techniques for calculating indefinite integrals. It is based on the product rule for differentiation and allows for the solving of integrals where the integrand is a product of two expressions that are not independently easy to integrate.

FORMULA FOR INTEGRATION BY PARTS

Let f(x)g(x) be the product of two differentiable functions. Then:
∫f(x)g′(x) dx = f(x)g(x) − ∫f′(x)g(x) dx.

In a more commonly used form, the formula is typically written as:

∫u dv = u * v − ∫v du,

where:

• u is the function that we differentiate (so du = u′ dx),
• dv is the function (or part of the integrand including dx) that we integrate (so v = ∫dv).

The success of the method relies on a smart choice of 'u' and 'dv' such that the integral ∫v du becomes simpler than the original integral ∫u dv.

CHOOSING FUNCTIONS U AND DV

For choosing the functions, the LIATE rule is often considered as a guideline (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential functions), where functions higher on the list are preferred choices for 'u'. This is because the derivatives of logarithmic and inverse trigonometric functions are algebraic, and algebraic functions become simpler when differentiated repeatedly. Exponential and trigonometric functions often cycle when differentiated or integrated.

EXAMPLE: ∫x * e^x dx

Choose:

• u = x → du = dx
• dv = e^x dx → v = ∫e^x dx = e^x

Apply the formula:
∫x * e^x dx = x * e^x − ∫e^x * dx
= x * e^x − e^x + C

Solution: ∫x * e^x dx = e^x(x − 1) + C.

EXAMPLE: ∫ln(x) dx

Here we use a special form where ln(x) is treated as 'u', and 'dv' is taken as 'dx'. We write ∫ln(x) dx as:
∫ln(x) * 1 dx.

Choose:

• u = ln(x) → du = (1/x) dx
• dv = 1 dx → v = ∫1 dx = x

It follows:
∫ln(x) dx = x * ln(x) − ∫x * (1/x) dx
= x * ln(x) − ∫1 dx
= x * ln(x) − x + C.

CONCLUSION

Integration by parts is a key method for solving integrals of products of functions. With the rule ∫u dv = u*v − ∫v du, we transform a more difficult integral into an easier one, where the success of the method depends on the correct choice of functions 'u' and 'dv'. This technique is often used multiple times in succession or in combination with other integration methods.