Calculus/Integration techniques/Irrational Functions

Integration techniques/Reduction Formula Integration techniques/Numerical Approximations Integration of irrational functions is more difficult than rational functions, and many cannot be done. However, there are some particular types that can be reduced to rational forms by suitable substitutions. Type 1 Integrand contains Use the substitution . Example Find . Find . Type 2 Integral is of the form Write as . Example Find . Type 3 Integrand contains , or This was discussed in "trigonometric substitutions above". Here is a summary: For , use . For , use . For , use . Type 4 Integral is of the form Use the substitution . Example Find . Type 5 Other rational expressions with the irrational function If , we can use . If , we can use . If can be factored as , we can use . If and can be factored as , we can use Integration techniques/Reduction Formula Integration techniques/Numerical Approximations