FoldUnfold Table of Contents Integration by Trigonometric Substitution Examples 2 Example 1 Example 2 Integration by Trigonometric Substitution Examples 2 We will now look at some more examples of integration by trigonometric substitution. Please look back at the Integration by Trigonometric Substitution Examples 1 for more examples. Additionally, recall the following table: Within the Integrand […]

## Integration by Trigonometric Substitution Examples 1

FoldUnfold Table of Contents Integration by Trigonometric Substitution Examples 1 Example 1 Example 2 Integration by Trigonometric Substitution Examples 1 We will now look at further examples of Integration by Trigonometric Substitution. For more examples, see the Integration by Trigonometric Substitution Examples 2 page. Also, recall the following table: Within the Integrand Appropriate Substitution Appropriate […]

## Integration by Trigonometric Substitution

FoldUnfold Table of Contents Integration by Trigonometric Substitution Example 1 Integration by Trigonometric Substitution Suppose that $f$ is a function and $a$ is a constant. Sometimes we come in contact with a function containing $\sqrt{a^2 – x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 – a^2}$. For example, consider evaluating the integral $\int \sqrt{a^2 + x^2} \: […]

## The Indefinite Integrals of Secant and Cosecant

FoldUnfold Table of Contents The Indefinite Integrals of Secant and Cosecant The Indefinite Integrals of Secant and Cosecant Evaluating $\int \sec x \: dx$ and $\int \csc x \: dx$ may seem difficult at first, but in actuality, it is not that difficult if we apply a special trick for these integrals. Theorem 1: The […]

## Advanced Trigonometric Function Integration 2

FoldUnfold Table of Contents Advanced Trigonometric Function Integration 2 Example 1 Trigonometric Products Containing Even Powers of Sine or Cosine. Example 2 Trigonometric Products of Tan and Secant where the Power of Secant is Even Example 3 Trigonometric Products of Tan and Secant where the Power of Tan is Odd Example 4 Advanced Trigonometric Function […]

## Advanced Trigonometric Function Integration Examples 1

FoldUnfold Table of Contents Advanced Trigonometric Function Integration Examples 1 Trigonometric Products Containing an Odd Power of Cosine Example 1 Example 2 Trigonometric Products Containing an Odd Power of Sine Example 3 Example 4 Trigonometric Products of Sin(ax)Cos(bx) Example 5 Advanced Trigonometric Function Integration Examples 1 We’re going to continue looking through examples of Advanced […]

## Advanced Trigonometric Function Integration

FoldUnfold Table of Contents Advanced Trigonometric Function Integration Advanced Trigonometric Function Integration So far we have integrated rather simple trigonometric functions such that $f(x) = \sin x$ where $\int f(x) \: dx = -\cos x$. We will now look at techniques for integrating more challenging trigonometric functions and prove the following theorem: Theorem 1: The […]

## Reduction Formulas

FoldUnfold Table of Contents Reduction Formulas Example 1 Example 2 Reduction Formulas Sometimes we may be interested in deriving a reduction formula for an integral, or a general identity for a seemingly complex integral. The list below outlines the most common reduction formulas: Reduction Formula for Sine: $\int \sin ^n x \: dx = -\frac{1}{n} […]

## Tabular Integration

FoldUnfold Table of Contents Tabular Integration Tabular Integration Type 1 Example 1 Tabular Integration Tabular integration is a special technique for integration by parts that can be applied to certain functions in the form $f(x) = g(x)h(x)$ where one of $g(x)$ or $h(x)$ is can be differentiated multiple times with ease, while the other function […]

## Integration by Parts of Definite Integrals

FoldUnfold Table of Contents Integration by Parts of Definite Integrals Example 1 Integration by Parts of Definite Integrals Suppose that we have a function that can integrated by parts, but we want to evaluate the integral at some lower bound $a$ and some upper bound $b$. Then it follows that: (1) \begin{align} \int_a^b u \: […]

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