How do you integrate #int dx/sqrt(16+x^2)^2# by trigonometric substitution?

Answer 1
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Assuming you meant: #intdx/sqrt(16+x^2)#
Let #x=4tantheta#. This implies that #dx=4sec^2thetacolor(white).d theta#. Substituting these in yields:
#=int(4sec^2thetacolor(white).d theta)/sqrt(16+16tan^2theta)=4/sqrt16intsec^2theta/sqrt(1+tan^2theta)d theta#
Recall that #1+tan^2theta=sec^2theta#:
#=intsec^2theta/secthetad theta=intsecthetacolor(white).d theta=lnabs(tantheta+sectheta)+C#
Rewriting in terms of tangent since our substitution is #tantheta=x/4#:
#=lnabs(tantheta+sqrt(1+tan^2theta))+C=lnabs(x/4+sqrt(1+x^2/16))+C#
#=lnabs(x/4+1/4sqrt(16+x^2))+C#
Factoring the #1/4# and moving it out of the integral as the constant #ln(1/4)# through the #log(AB)=log(A)+log(B)# rule, we see it combines with the constant of integration:
#=lnabs(x+sqrt(16+x^2))+C#
#" "#
Assuming you meant: #intdx/(sqrt(16+x^2))^2#

The square root and the exponent cancel, leaving just:

#=intdx/(16+x^2)#
Now use the same substitution as before, #x=4tantheta# such that #dx=4sec^2thetacolor(white).d theta#.
#=int(4sec^2thetacolor(white).d theta)/(16+16tan^2theta)=1/4intsec^2theta/(1+tan^2theta)d theta=1/4intd theta#
#=1/4theta+C#
From the substitution #x=4tantheta# solving for #theta# yields:
#=1/4arctan(x/4)+C#
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Answer 2

To integrate ( \int \frac{dx}{\sqrt{16+x^2}^2} ) using trigonometric substitution, we can let ( x = 4\tan(\theta) ), where ( \theta ) is the trigonometric angle.

Then we find ( dx ) by differentiating ( x = 4\tan(\theta) ) with respect to ( \theta ).

Next, we express ( \sqrt{16+x^2} ) in terms of ( \tan(\theta) ) using the trigonometric identity ( \sec^2(\theta) = 1 + \tan^2(\theta) ).

After that, we substitute ( x ) and ( dx ) in the integral with expressions involving ( \theta ).

We simplify the integrand using the substitution and then integrate with respect to ( \theta ).

Finally, we substitute back ( x ) in terms of ( \theta ) to express the result in terms of ( x ).

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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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