Chapter 6: Techniques of Integration
Section 6.1: Integration by Parts
Table 6.1.1 lists several forms of the integration-by-parts algorithm, a useful technique for obtaining antiderivatives.
∫fx⋅g′x ⅆx=fxgx−∫f′x⋅gx ⅆx
∫u dv=u v−∫v ⅆu
∫abfx⋅g′x ⅆx=fxgxx=ax=b−∫abf′x⋅gx ⅆx
∫abu ⅆv=u vab−∫abv ⅆu
Table 6.1.1 Various forms of the integration-by-parts algorithm
The first row of Table 6.1.1 provides two forms for applying parts integration to an indefinite integral; the second row, two forms for definite integration. Note that for the definite integral, the "boundary term" u v must also be evaluated at the endpoints so that u vab=ubvb−uava.
One of Maple's built-in tools uses the form in the left-hand column; the form in the right-hand column is probably easier to remember.
The algorithm is obtained by integrating the product rule for differentiation:
=∫u′x⋅vx ⅆx+∫ux⋅v′x ⅆx
∫ux⋅v′x ⅆx=uxvx−∫u′x⋅vx ⅆx.
∫u dv=u v−∫v ⅆu
Using the technique of integration by parts, evaluate ∫x sinx ⅆx.
Using the technique of integration by parts, evaluate ∫lnx ⅆx.
Using the technique of integration by parts, evaluate the definite integral ∫01tan−1x ⅆx.
Use integration by parts to establish the formula
∫ea xcosb x ⅆx = ea xa2+b2a cosb x+b sinb x.
Use integration by parts to evaluate the indefinite integral ∫x2sinx ⅆx.
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