121616949-math.184

# 121616949-math.184 - high degree polynomials in the...

This preview shows page 1. Sign up to view the full content.

170 Chapter 8 Techniques of Integration Exercises Find the antiderivatives. 1. Z x cos x dx 2. Z x 2 cos x dx 3. Z xe x dx 4. Z xe x 2 dx 5. Z sin 2 x dx 6. Z ln x dx 7. Z x arctan x dx 8. Z x 2 sin x dx 9. Z x sin 2 x dx 10. Z x sin x cos x dx 11. Z arctan( x ) dx 12. Z sin( x ) dx 13. Z sec 2 x csc 2 x dx A rational function is a fraction with polynomials in the numerator and denominator. For example, x 3 x 2 + x - 6 , 1 ( x - 3) 2 , x 2 + 1 x 2 - 1 , are all rational functions of x . There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function. The algebraic steps in that technique are rather cumbersome if the polynomial in the denominator has degree more than 2, and the technique requires that we factor the denominator, something that is not always possible. However, in practice one does not often run across rational functions with
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: high degree polynomials in the denominator for which one has to ﬁnd the antiderivative function. So we shall explain how to ﬁnd the antiderivative of a rational function only when the denominator is a quadratic polynomial ax 2 + bx + c . We should mention a special type of rational function that we already know how to integrate: If the denominator has the form ( ax + b ) n , the substitution u = ax + b will always work. The denominator becomes u n , and each x in the numerator is replaced by ( u-b ) /a , and dx = du/a . While it may be tedious to complete the integration if the numerator has high degree, it is merely a matter of algebra....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern