How do you integrate #int sqrt(-x^2-10x)/xdx# using trigonometric substitution?

Answer 1

Use the substitution #x+5=5sintheta#.

Let

#I=intsqrt(-x^2-10x)/xdx#

Complete the square in the square root:

#I=intsqrt(25-(x+5)^2)/xdx#
Apply the substitution #x+5=5sintheta#:
#I=int(5costheta)/(5sintheta-5)(5costhetad theta)#

Simplify:

#I=5intcos^2theta/(sintheta-1)d theta#
Apply the identity #sin^2theta+cos^2theta=1#:
#I=5int(1-sin^2theta)/(sintheta-1)d theta#
Apply the difference of squares #a^2-b^2=(a-b)(a+b)#:
#I=-5int(1+sintheta)d theta#

Integrate directly:

#I=-5(theta-costheta)+C#

Reverse the substitution:

#I=sqrt(25-(x+5)^2)-5sin^(-1)((x+5)/5)+C#
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Answer 2

The answer is #=-5arcsin(1/5(x+5))+5sqrt(1-((x+5)/5)^2)+C#

Complete the square :

#-x^2-10x=25-(x+5)^2#

Therefore, the integral is

#I=int(sqrt(-x^2-10x)dx)/(x)=int(sqrt(25-(x+5)^2)dx)/x#
Let #u=x+5#, #=>#, #du=dx#
#I=int(sqrt(25-u^2)du)/(u-5)#
Let #u=5sinv#, #=>#, #du=5cosvdv#

Therefore,

#I=int((5cosv)*5cosvdv)/(5(sinv-1))#
#=5int(cos^2vdv)/(-(1-sinv))#
#=-5int(1+sinv)dv#
#=-5v+5cosv#
#=-5arcsin(u/5)+5sqrt(1-(u/5)^2)#
#=-5arcsin(1/5(x+5))+5sqrt(1-((x+5)/5)^2)+C#
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Answer 3

To integrate ( \int \frac{\sqrt{-x^2 - 10x}}{x} , dx ) using trigonometric substitution, let ( x = -5 \sin(\theta) - 5 ). Then, ( dx = -5 \cos(\theta) , d\theta ). Substitute these expressions into the integral and simplify it using trigonometric identities.

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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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