Explanation
Integration by parts is a method to integrate products of functions. It arises from reversing the product rule in differential calculus. The formula for integration by parts is:
Procedure
To integrate products, first cleverly choose u(x) and v'(x). Here, u(x) should become as simple as possible when differentiated, and v'(x) should be easy to integrate. Then determine u'(x) and v(x) and plug these into the integration by parts formula.
Examples
Calculate \int x \cdot e^{x} \,dx.
Choose u(x)=x and v'(x)=e^{x}. Then it is u'(x)=1 and v(x)=e^{x}. Substitution yields: \int x \cdot e^{x} \,dx = x \cdot e^{x} - \int 1 \cdot e^{x} \,dx = x \cdot e^{x} - e^{x} + C.
Calculate \int x^{2} \cdot \sin(x) \,dx.
Choose u(x)=x^{2} and v'(x)=\sin(x). Then it is u'(x)=2x and v(x)=-\cos(x). Substitution yields: \int x^{2}\sin(x)dx = -x^{2}\cos(x) + \int 2x\cos(x)dx. Now apply integration by parts again with u(x)=2x and v'(x)=\cos(x). Result: -x^{2}\cos(x) + 2(x\sin(x)+\cos(x)) + C.
Note
- ★Partial integration is the inverse of the product rule.
- ★Choose u(x) so that it is easier to differentiate, and v'(x) so that it is easily integrated.
Exercises
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