Integral calculus
The indefinite integral Integration by parts rule
Integration by parts rule
The rule for differentiating the product of two differentiable functions leads to the integration by parts formula.
Let  f (x) and g (x) are differentiable functions, then the product rule gives
[ f (x) g (x)]' =  f (x) g (x)'  +  g (x) f ' (x),
by integrating both sides Since the integral of the derivative of a function is the function itself, then and by rearranging obtained is the integration by parts formula.
By substituting  u = f (x) and  v = g (x) then,  du = f ' (x) dx and  dv = g' (x) dx, so that To apply the above formula, the integrand of a given integral should represent the product of one function and
the differential of the other.
The selection of the function u and the differential dv should simplify the evaluation of the remaining integral.
In some cases it will be necessary to apply the integration by parts repeatedly to obtain a simpler integral.
Evaluating the indefinite integrals using the integration by parts formula, examples
Evaluate the following indefinite integrals using the integration by parts formula;
 41 42 43 44 45 46 Evaluating the indefinite integrals using the integration by parts formula, solutions
 Example:   41.  Evaluate Solution: Example:   42.  Evaluate Solution: Example:   43. Evaluate Solution: Example:   44. Evaluate Solution: Example:   45. Evaluate Solution: Example:   46. Evaluate Solution:    Calculus contents E 