Explanation Consider the integral containing the integrand function a 2 − x 2 . Write the general form of such an integral as below. ∫ d x a 2 − x 2 In this case, assume the variable x as a sin θ . x = a sin θ Take the inverse of the function of the above substitution as below. θ = sin − 1 ( x a ) Here, the inverse will be valid and present only when the value of angle θ has to take the values between − π 2 and π 2 with the limits also included. − π 2 ≤ θ ≤ π 2 Consider the integral containing the integrand function a 2 + x 2 . Write the general form of such an integral as below. ∫ d x a 2 + x 2 In this case, assume the variable x as a tan θ . x = a tan θ Take the inverse of the function of the above substitution as below. θ = tan − 1 ( x a ) Here, the inverse will be valid and present only when the value of angle θ has to take the values between − π 2 and π 2 with the limits not included. − π 2 < θ < π 2 Consider the integral containing the integrand function x 2 − a 2 . Write the general form of such an integral as below. ∫ d x x 2 − a 2 In this case, assume the variable x as a sec θ . x = a sec θ Take the inverse of the function of the above substitution as below. θ = sec − 1 ( x a ) The above function x = a sec θ is reversible if the value of x a is greater than or equal to 1, in case of the limits for the angle θ lies between 0 and π 2 with the value of 0 also included

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Author: WEIR

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Chapter 8, Problem 6GYR

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What are the restrictions to be placed on the variables involved in the three basic trigonometric substitutions in order to make sure that these substitutions are reversible and can have inverse of it.

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Chapter 8, Problem 5GYR

Chapter 8, Problem 7GYR

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