Derivative and Integral of Inverse Sine
In this section we will discuss about the derivative and integral of the inverse sine function.
Derivative of Inverse Sine
The derivative of the inverse sine function, denoted as (y = sin-1x), is found by considering the relationship (sin y = x). Using the chain rule, the derivative is expressed as:
dy/dx = 1/cos y
Further, by using the Pythagorean identity (sin2y + cos2y = 1) the expression for cos y is substituted:
(dy / dx) = 1 / √(1 – x2)
Therefore, the derivative of the inverse sine function, or sin-1x, is 1 / √ (1 – x2)
Integral of Inverse Sine
Find the integral of sin-1x with respect to (x) using the integration by parts method. To start, we express the integral as:
∫ sin-1x. 1 dx
Following the LIATE rule, where f(x) = sin-1x and g(x) = 1, the integration by parts formula is applied:
∫ f(x). g(x) dx = f(x) ∫ g(x)dx – ∫ [{(d/dx) f(x)}. ∫ g(x) dx] dx + C
Substituting f(x) = sin-1x and g(x) = 1, we get:
∫ f(x). g(x) dx = f(x) ∫ g(x)dx – ∫ [{(d/dx) f(x)}. ∫ g(x) dx] dx + C
Now, we evaluate the inner integral using the u-substitution method. Let (u = 1 – x2), then (-2xdx = du) (or x dx = -1/2 du). Substituting these, we get:
∫sin-1 dx = xsin-1x + ∫ (1 / √u)(-1/2) du}+ C
∫sin-1 dx = xsin-1x – √u+ C
Replace (u) with 1-x2, we get:
∫sin-1 dx = xsin-1x + √(1 – x2) + C
Therefore, the integral of sin-1x with respect to (x) is xsin-1x + √(1 – x2) + C
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Inverse Sine
Inverse Sine function is one of the important inverse trigonometric functions. The inverse of trigonometric functions gives the angle with the help of the sides of the triangles. In this article, we will explore the inverse sine function, its definition, formula, and properties. We will also discuss the domain and range of the inverse sine function, the graph of the inverse sine function, the derivative of inverse sine, and the integral of inverse sine along with the solved examples. Let’s start our learning on the topic of “Inverse Sine”.
Table of Content
- What is Sine Function?
- What is Inverse Sine Function?
- Properties of Inverse Sine Function
- Domain and Range of Inverse Sine
- Graph of Inverse Sine Function
- Derivative and Integral of Inverse Sine
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