How do you find the integral of #int 5/sqrt(9-x^2)dx#?

Answer 1
Let #x=3*sint# hence #dx=3cost*dt# then we have that
#int 5/sqrt(9-(3sint)^2)*3cost*dt#
#int 5/sqrt(9-9*sin^2t)*3cost*dt#
#int 5/(sqrt9*sqrt(1-sin^2t))*3*cost*dt#
#int 5/(3*sqrt(cos^2t))*3*cost*dt#
#int 5/(3*cost)*3*cost*dt#
#int 5/(cancel3*cancelcost)*cancel3*cancelcost*dt#
#int 5dt=5*t+c#
But hence #sint=x/3# or #t=arcsin(x/3)#

We have that

#int 5/(sqrt(9-x^2))*dx=5*arcsin(x/3)+c#
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Answer 2

Please see the explanation.

You can find the general form of the integral in any list of integrals:

#int1/sqrt(a^2 - x^2)dx = sin^-1(x/a) + C#

However, I think you want to be shown how to do the trigonometric substitution that put this general form on the list. Let's begin:

Given: #int5/sqrt(9 - x^2)dx = ?#
Let #x = 3sin(theta)#, then #dx = 3cos(theta)d"theta#

Substitute the above into the integral:

#5int(3cos(theta))/sqrt(9 - 9sin^2(theta))d"theta = #

Factor 9 from under the radical:

#5int(3cos(theta))/(3sqrt(1 - sin^2(theta)))d"theta = #
Use the identity #cos(theta) = sqrt(1 - sin^2(theta)#:
#5int(3cos(theta))/(3cos(theta))d"theta = #

The numerator and denominator cancel to become 1:

#5intd"theta = 5theta + C#
To reverse the substitution, we need to describe #theta# in terms of x so we start with equation that we used to make the substution:
#x = 3sin(theta)#
#x/3 = sin(theta)#
#theta = sin^-1(x/3); -3 <=x <= 3#
#int5/sqrt(9 - x^2)dx = 5sin^-1(x/3) + C; -3 <=x <= 3#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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