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| Integral
calculus |
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The
indefinite integral |
Integration by parts rule |
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Evaluating the indefinite integrals using the integration by parts
formula, examples
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Evaluating the indefinite integrals using the integration by parts
formula, solutions
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Integration
by parts rule
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The rule for differentiating the
product of two differentiable functions leads to the integration
by parts formula.
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Let f
(x)
and g
(x)
are differentiable functions, then the product rule gives
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[ f
(x) g (x)]'
= f (x)
g (x)' + g (x)
f ' (x),
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integrating both sides |
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Since the integral of the derivative
of a function is the function itself, then
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and by rearranging obtained is
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the integration by parts
formula.
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By substituting
u = f
(x) and
v
= g (x)
then, du =
f ' (x)
dx and dv
= g' (x)
dx, so that
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To apply the above formula, the
integrand of a given integral should represent the product of one
function and the differential of the other.
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The selection of the function
u and the differential dv
should simplify the
evaluation of the remaining integral.
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In some cases it will be necessary to
apply the integration by parts repeatedly to obtain a simpler
integral.
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| Evaluating the indefinite integrals using the integration by parts
formula, examples
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| Evaluate
the following indefinite integrals using the integration by parts
formula;
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48. |
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50. |
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| 51. |
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52. |
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| Evaluating the indefinite integrals using the integration by parts
formula, solutions
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| Example:
47. Evaluate |
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| Example:
48.
Evaluate |
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| Example:
49.
Evaluate |
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| Example:
50.
Evaluate |
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| Example:
51.
Evaluate |
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| Example:
52. Evaluate |
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| Calculus contents
E |
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