- #1
NotaMathPerson
- 83
- 0
$$\int(1-y^2)^\frac{1}{2}\,dy$$
I did trig substitution
$$y=\sin\theta$$
$$dy=\cos\theta\,d\theta$$
$$\int(1+\cos2\theta)d\theta$$
$$\arcsin\,y+\frac{1}{2}\sin(2\arcsin\,y)+c$$
How do I get rid of the arcsins?
The formula for integrating $$1-y^2$$ is $$\int (1-y^2)dy = y - \frac{y^3}{3} + C$$, where C is the constant of integration.
Simplifying arcsins is important because it allows us to express the solution in a simpler and more concise form. It also helps us to better understand the relationship between the given function and its inverse trigonometric function.
To simplify arcsins of 1-y^2, we can use the trigonometric identity $$\sin^2x + \cos^2x = 1$$, where x is the angle. Rearranging this identity, we get $$\sin^2x = 1-\cos^2x$$. Substituting x with arcsin(y), we get $$\sin^2(arcsin(y)) = 1- \cos^2(arcsin(y)) = 1-y^2$$. Therefore, $$arcsin(y) = \sqrt{1-y^2}$$.
Yes, we can integrate 1-y^2 without simplifying arcsins. However, the resulting integral will be more complex and may require more steps to solve. Simplifying arcsins can make the integration process easier and more manageable.
Integrating 1-y^2 has various applications in physics, engineering, and other scientific fields. For example, it can be used to calculate the area under a curve, which is important in finding the work done by a variable force. It is also useful in finding the volume of a solid of revolution and in solving problems involving projectile motion.