Trigonometric Integrals, Trigonometric Substitution, and Partial Fractions - The Questions - 1,001 Calculus Practice Problems

1,001 Calculus Practice Problems

Part I

The Questions

Chapter 14

Trigonometric Integrals, Trigonometric Substitution, and Partial Fractions

This chapter covers trigonometric integrals, trigonometric substitutions, and partial fractions — the remaining integration techniques you encounter in a second-semester calculus course (in addition to u-substitution and integration by parts; see Chapter 13). In a sense, these techniques are nothing fancy. For the trigonometric integrals, you typically use a u-substitution followed by a trigonometric identity, possibly throwing in a bit of algebra to solve the problem. For the trigonometric substitutions, you're often integrating a function involving a radical; by picking a clever substitution, you can often remove the radical and make the problem into a trigonometric integral and proceed from there. Last, the partial fractions technique simply decomposes a rational function into a bunch of simple fractions that are easier to integrate.

With that said, many of these problems have many steps and require you to know identities, polynomial long division, derivative formulas, and more. Many of these problems test your algebra and trigonometry skills as much as your calculus skills.

The Problems You'll Work On

This chapter finishes off the integration techniques that you see in a calculus class:

· Solving definite and indefinite integrals involving powers of trigonometric functions

· Solving definite and indefinite integrals using trigonometric substitutions

· Solving definite and indefinite integrals using partial fraction decompositions

What to Watch Out For

You can get tripped up in a lot of little places on these problems, but hopefully these tips will help:

· Not all of the trigonometric integrals fit into a nice mold. Try identities, u-substitutions, and simplifying the integral if you get stuck.

· You may have to use trigonometry and right triangles in the trigonometric substitution problems to recover the original variable.

· If you've forgotten how to do polynomial long division, you can find some examples in Chapter 1's algebra review.

· The trigonometric substitution problems turn into trigonometric integral problems, so make sure you can solve a variety of the latter problems!

Trigonometric Integrals

884–913 Find the antiderivative or evaluate the definite integral.

884. 9781118496718-eq14001.eps

885. 9781118496718-eq14002.eps

886. 9781118496718-eq14003.eps

887. 9781118496718-eq14004.eps

888. 9781118496718-eq14005.eps

889. 9781118496718-eq14006.eps

890. 9781118496718-eq14007.eps

891. 9781118496718-eq14008.eps

892. 9781118496718-eq14009.eps

893. 9781118496718-eq14010.eps

894. 9781118496718-eq14011.eps

895. 9781118496718-eq14012.eps

896. 9781118496718-eq14013.eps

897. 9781118496718-eq14014.eps

898. 9781118496718-eq14015.eps

899. 9781118496718-eq14016.eps

900. 9781118496718-eq14017.eps

901. 9781118496718-eq14018.eps

902. 9781118496718-eq14019.eps

903. 9781118496718-eq14020.eps

904. 9781118496718-eq14021.eps

905. 9781118496718-eq14022.eps

906. 9781118496718-eq14023.eps

907. 9781118496718-eq14024.eps

908. 9781118496718-eq14025.eps

909. 9781118496718-eq14026.eps

910. 9781118496718-eq14027.eps

911. 9781118496718-eq14028.eps

912. 9781118496718-eq14029.eps

913. 9781118496718-eq14030.eps

Trigonometric Substitutions

914–939 Evaluate the integral using a trigonometric substitution.

914. 9781118496718-eq14031.eps

915. 9781118496718-eq14032.eps

916. 9781118496718-eq14033.eps

917. 9781118496718-eq14034.eps

918. 9781118496718-eq14035.eps

919. 9781118496718-eq14036.eps

920. 9781118496718-eq14037.eps

921. 9781118496718-eq14038.eps

922. 9781118496718-eq14039.eps

923. 9781118496718-eq14040.eps

924. 9781118496718-eq14041.eps

925. 9781118496718-eq14042.eps

926. 9781118496718-eq14043.eps

927. 9781118496718-eq14044.eps

928. 9781118496718-eq14045.eps

929. 9781118496718-eq14046.eps

930. 9781118496718-eq14047.eps

931. 9781118496718-eq14048.eps

932. 9781118496718-eq14049.eps

933. 9781118496718-eq14050.eps

934. 9781118496718-eq14051.eps

935. 9781118496718-eq14052.eps

936. 9781118496718-eq14053.eps

937. 9781118496718-eq14054.eps

938. 9781118496718-eq14055.eps

939. 9781118496718-eq14056.eps

Finding Partial Fraction Decompositions (without Coefficients)

940–944 Find the partial fraction decomposition without finding the coefficients.

940. 9781118496718-eq14057.eps

941. 9781118496718-eq14058.eps

942. 9781118496718-eq14059.eps

943. 9781118496718-eq14060.eps

944. 9781118496718-eq14061.eps

Finding Partial Fraction Decompositions (Including Coefficients)

945–949 Find the partial fraction decomposition, including the coefficients.

945. 9781118496718-eq14062.eps

946. 9781118496718-eq14063.eps

947. 9781118496718-eq14064.eps

948. 9781118496718-eq14065.eps

949. 9781118496718-eq14066.eps

Integrals Involving Partial Fractions

950–958 Evaluate the integral using partial fractions.

950. 9781118496718-eq14067.eps

951. 9781118496718-eq14068.eps

952. 9781118496718-eq14069.eps

953. 9781118496718-eq14070.eps

954. 9781118496718-eq14071.eps

955. 9781118496718-eq14072.eps

956. 9781118496718-eq14073.eps

957. 9781118496718-eq14074.eps

958. 9781118496718-eq14075.eps

Rationalizing Substitutions

959–963 Use a rationalizing substitution and partial fractions to evaluate the integral.

959. 9781118496718-eq14076.eps

960. 9781118496718-eq14077.eps

961. 9781118496718-eq14078.eps

962. 9781118496718-eq14079.eps

963. 9781118496718-eq14080.eps