MAIN METHODS OF INTEGRATIONIntegration 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(х) - 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
|
.
. Notice that the numerator is the derivative of the denominator
. Differentiating gives 
and hence 
.
we have shown that
.
