Resources for Integrations that give Inverse Trig Functions
-
Video Tutorials
3
Click Here
Integrations that give Inverse Trig Functions Theory
![The following rules may he applied \\ $\begin{aligned} & \displaystyle \int \dfrac{1}{\sqrt{1-x^2}} d x=\sin ^{-1} x+c \\ & \displaystyle \int \dfrac{1}{\sqrt{a^2-x^2}} d x=\sin ^{-1} \dfrac{x}{a}+c \\ & \displaystyle \int \dfrac{-1}{\sqrt{1-x^2}} d x=\cos ^{-1} x+c \\ & \displaystyle \int \dfrac{-1}{\sqrt{a^2-x^2}} d x=\cos ^{-1} \dfrac{x}{a}+c \\ & \displaystyle \int \dfrac{1}{1+x^2} d x=\tan ^{-1} x+c \\ & \displaystyle \int \dfrac{1}{a^2+x^2} d x=\dfrac{1}{a} \tan ^{-1} \dfrac{x}{a}+c . \end{aligned}$\\ \begin{multicols}{2} \textbf{Example 1}\\ %question 123221 Evaluate \(\displaystyle \int_{-1}^1 \dfrac{1}{\sqrt{2-x^2}}\, d x\) \\ \textbf{Example 1 solution}\\ $\begin{aligned} \displaystyle \int_{-1}^1 \frac{1}{\sqrt{2-x^2}} d x &=\int_{-1}^1 \frac{1}{\sqrt{(\sqrt{2})^2-x^2}} d x \\ &=\left[\sin ^{-1} \frac{x}{\sqrt{2}}\right]_{-1}^1 \\ &=\left[\sin ^{-1} \frac{1}{\sqrt{2}}-\sin ^{-1} \frac{-1}{\sqrt{2}}\right] \\ &=\left[\sin ^{-1} \frac{1}{\sqrt{2}}+\sin ^{-1} \frac{1}{\sqrt{2}}\right] \\ &=2 \sin ^{-1} \frac{1}{\sqrt{2}} \\ &=2 \times \frac{\pi}{4} \\ &=\frac{\pi}{2} \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %question 123223 Evaluate \(\displaystyle \int_0^{\frac{1}{3}} \dfrac{1}{1+9 x^2}\, d x\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \displaystyle \int_0^{\frac{1}{3}} \frac{1}{1+9 x^2} d x &=\int_0^{\frac{1}{3}} \frac{1}{9\left(\frac{1}{9}+x^2\right)} d x \\ &=\frac{1}{9} \int_0^{\frac{1}{3}} \frac{1}{\left(\frac{1}{3}\right)^2+x^2} d x \\ &=\frac{1}{9}\left[3 \tan ^{-1} 3 x\right]_0^{\frac{1}{3}} \\ &=\frac{1}{3}\left[\tan ^{-1}(1)-\tan ^{-1}(0)\right] \\ &=\frac{1}{3} \times \frac{\pi}{4} \\ &=\frac{\pi}{12} \end{aligned}$\\ \end{multicols}](/media/azuffa4b/integrations-that-give-inverse-trig-functions.png)
NSW Y12 Maths - Extension 1 Calculus Integrations that give Inverse Trig Functions
Videos
Videos relating to Integrations that give Inverse Trig Functions.
-
Integrations that give Inverse Trig Functions - Video - Lesson
-
Integrations that give Inverse Trig Functions - Video - Exercises
-
Integrations that give Inverse Trig Functions - Video - Integration Resulting in Inverse trigonometric functions
Plans & Pricing
With all subscriptions, you will receive the below benefits and unlock all answers and fully worked solutions.
-
Teachers TutorsFeaturesFreeProAll ContentAll courses, all topicsQuestionsAnswersWorked SolutionsSystemYour own personal portalQuizbuilderClass ResultsStudent ResultsExam RevisionRevision by TopicPractise ExamsAnswersWorked SolutionsAwesome StudentsFeaturesFreeProContentAny course, any topicQuestionsAnswersWorked SolutionsSystemYour own personal portalBasic ResultsAnalyticsStudy RecommendationsExam RevisionRevision by TopicPractise ExamsAnswersWorked Solutions
Books / e-books
Course Book
Purchase the course book. This book either comes as a physical book or it can be purchased as an e-book.
Topic Books
You may choose to purchase the individual topic book from the main coursebook. These only come as e-books.
