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 . Notice that the numerator is the derivative of the denominator
Let . Differentiating gives and hence .
Substituting this change of variable the integral becomes
Now by expressing this result in terms of we have shown that
Integration by parts
By the Product Rule for Derivatives, . Thus,
. This formula for integration
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 and and . Thus,
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: 760