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MAIN METHODS OF INTEGRATION
Integration by substitution ( or change of variable )
Steps for Integrating by Substitution—Indefinite Integrals: 1. Choose a substitution u = g(x), such as the inner part of a composite function. 2. Compute 3. Re-write the integral in terms of u and du. 4. Find the resulting integral in terms of u. 5. Substitute g(x) back in for u, yielding a function in terms of x only. 6. Check by differentiating.
If f(x) is continous function, F(x)- its antiderivative and φ(ő)- differentiable function, then In the particular case Example 8. To find Let Substituting this change of variable the integral becomes Now by expressing this result in terms of
Integration by parts By the Product Rule for Derivatives,
by parts often makes it possible to reduce a complicated integral involving a product to
a simpler integral. By letting we get the more common formula for integration by parts: Example 9. Find
Let
It is possible that when you set up an integral using integration by parts, the resulting integral will be more complicated than the original integral. In this case, change your substitutions for u and dv.
Date: 2015-01-02; view: 2100 |