INTEGRATION BY PARTS FORMULA EXPLAINED

Integration by parts is a technique used in calculus to evaluate integrals of the form ∫u(x)dv(x) or ∫v(x)du(x).

The basic idea behind the method is to use the product rule of differentiation, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. This rule can be rearranged and integrated to give the integration by parts formula:

∫u(x)dv(x) = u(x)v(x) - ∫v(x)du(x)

The formula is used by choosing two functions, u(x) and v(x), such that their derivatives can be easily evaluated. The integral of the product of u(x) and the derivative of v(x) is then calculated and subtracted from the product of v(x) and the derivative of u(x).

The integration by parts formula can be applied to a wide variety of integrals. 

For example:

if we want to evaluate the integral of xe^x, 

we can choose u(x) = x and v'(x) = e^x

The derivative of u(x) is then 1 and the integral of v'(x) is v(x) = e^x. Substituting these into the formula gives:

∫xe^x dx = xe^x - ∫e^x dx

The integral on the right side can be easily evaluated, giving e^x. So the final solution is:

∫xe^x dx = xe^x - e^x + C

Another example is the integral of ln(x)sin(x). 

In this case, we can choose u(x) = ln(x) and v'(x) = sin(x).

The derivative of u(x) is 1/x and the integral of v'(x) is v(x) = -cos(x). Substituting these into the formula gives:

∫ln(x)sin(x) dx = ln(x)(-cos(x)) - ∫(-cos(x))(1/x) dx

The integral on the right side can be evaluated using integration by substitution, giving:

ln(x)sin(x) + cos(x)ln(x) + C.

Note : It should be noted that integration by parts is not always possible or practical. In some cases, the integral may not converge or may be too difficult to evaluate. In such cases, other techniques, such as substitution or partial fractions, may need to be used. Additionally, it is important to choose the right functions for u(x) and v(x) in order to make the calculation as simple as possible.

In summary, integration by parts is a powerful technique for evaluating integrals of the form ∫u(x)dv(x) or ∫v(x)du(x). It makes use of the product rule of differentiation and can be applied to a wide variety of integrals. However, it is not always possible or practical to use and other techniques may need to be employed. Choosing the right functions for u(x) and v(x) is also important for making the calculation as simple as possible.