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17Calculus Integrals - Trig Integration For Calculus 2

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Trig Integration For Calculus 2

This page is for calculus 2 students who have mastered the basics of trig integration in calculus 1, which are covered on a separate tutorial page.
On this current page, we briefly describe more trig integration techniques and provide links to our tutorials for those techniques.

Topics You Need To Understand For This Page

[basics of integration] - [trig integration for calculus 1] - [integration by substitution]

Difference Between Trig Integration and Trig Substitution

Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand.

Trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

Basic Trigonometric Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Basic Trigonometric Integration (From Calculus 1)

When you are first starting to learn integration, you will run across problems involving trig functions. The basic information and techniques you need are the basic trig identities and integration by substitution. In addition to the identities in the table above, you need to know these integrals.

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

Additional Strategies For Calculus 2

Here are links to tutorials and practice problems involving more advanced techniques, usually found in second semester calculus.

integrands involving sine and/or cosine

integrands involving secant and/or tangent

A unique, more specialized strategy is called the Weierstrass substitution.

The strategies listed so far on this page and the trig integration for calculus 1 page cover most of the integrals you will run across. Occasionally, you may need various other techniques to convert the integrand into a form that can be integrated. Here are a few ideas.
1. Convert all the trig functions into sine and cosine. Sometimes, you may have lots of cancellation and end up with an easy integral.
2. Use the half-angle formulas to remove powers. This may leave an easy substitution problem.

Once you get more experience, you will be able to tell pretty quickly which strategy is best.

Okay, so you are ready to test yourself with some practice problems. You will find lots of practice problems on the special dedicated page to calculus 2 integrals.

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