1,001 Calculus Practice Problems

Part I

The Questions

 

Chapter 13

U-Substitution and Integration by Parts

In this chapter, you encounter some of the more advanced integration techniques: u-substitution and integration by parts. You use u-substitution very, very often in integration problems. For many integration problems, consider starting with a u-substitution if you don't immediately know the antiderivative. Another common technique is integration by parts, which comes from the product rule for derivatives. One of the difficult things about these problems is that even when you know which procedure to use, you still have some freedom in how to proceed; what to do isn't always clear, so dive in and try different things.

The Problems You'll Work On

This chapter is the start of more challenging integration problems. You work on the following skills:

·        Using u-substitution to find definite and indefinite integrals

·        Using integration by parts to find definite and indefinite integrals

What to Watch Out For

Here are a few things to keep in mind while working on the problems in this chapter:

·        Even if you know you should use a substitution, there may be different substitutions to try. As a rule, start simple and make your substitution more complex if your first choice doesn't work.

·        When using a u-substitution, don't forget to calculate du, the differential.

·        You can algebraically manipulate both du and the original u-substitution, so play with both!

·        For the integration by parts problems, if your pick of u and dv don't seem to be working, try switching them.

Using u-Substitutions

832–857 Use substitution to evaluate the integral.

832. 9781118496718-eq13001.eps

833. 9781118496718-eq13002.eps

834. 9781118496718-eq13003.eps

835. 9781118496718-eq13004.eps

836. 9781118496718-eq13005.eps

837. 9781118496718-eq13006.eps

838. 9781118496718-eq13007.eps

839. 9781118496718-eq13008.eps

840. 9781118496718-eq13009.eps

841. 9781118496718-eq13010.eps

842. 9781118496718-eq13011.eps

843. 9781118496718-eq13012.eps

844. 9781118496718-eq13013.eps

845. 9781118496718-eq13014.eps

846. 9781118496718-eq13015.eps

847. 9781118496718-eq13016.eps

848. 9781118496718-eq13017.eps

849. 9781118496718-eq13018.eps

850. 9781118496718-eq13019.eps

851. 9781118496718-eq13020.eps

852. 9781118496718-eq13021.eps

853. 9781118496718-eq13022.eps

854. 9781118496718-eq13023.eps

855. 9781118496718-eq13024.eps

856. 9781118496718-eq13025.eps

857. 9781118496718-eq13026.eps

Using Integration by Parts

858–883 Use integration by parts to evaluate the integral.

858. 9781118496718-eq13027.eps

859. 9781118496718-eq13028.eps

860. 9781118496718-eq13029.eps

861. 9781118496718-eq13030.eps

862. 9781118496718-eq13031.eps

863. 9781118496718-eq13032.eps

864. 9781118496718-eq13033.eps

865. 9781118496718-eq13034.eps

866. 9781118496718-eq13035.eps

867. 9781118496718-eq13036.eps

868. 9781118496718-eq13037.eps

869. 9781118496718-eq13038.eps

870. 9781118496718-eq13039.eps

871. 9781118496718-eq13040.eps

872. 9781118496718-eq13041.eps

873. 9781118496718-eq13042.eps

874. 9781118496718-eq13043.eps

875. 9781118496718-eq13044.eps

876. 9781118496718-eq13045.eps

877. 9781118496718-eq13046.eps, where m ≠ 0

878. 9781118496718-eq13047.eps

879. 9781118496718-eq13048.eps

880. 9781118496718-eq13049.eps

881. 9781118496718-eq13050.eps

882. 9781118496718-eq13051.eps

883. 9781118496718-eq13052.eps