Integral calculus
The indefinite integral

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;
 47 48
 49 50
 51 52
Evaluating the indefinite integrals using the integration by parts formula, solutions
 Example:   47. Evaluate
 Solution:
 Example:   48.  Evaluate
 Solution:
 Example:   49.  Evaluate
 Solution:
 Example:   50.  Evaluate
 Solution:
 Example:   51.  Evaluate
 Solution:
 Example:   52. Evaluate
 Solution:
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