How do you integrate #int sec^2x/(4-tan^2x)^(3/2)# by trigonometric substitution?

Answer 1

#tanx/(4sqrt(4-tan^2x))+C#

We have:

#I=intsec^2x/(4-tan^2x)^(3/2)dx#
We will first use the non-trigonometric substitution #u=tanx#, implying that #du=sec^2xdx#:
#I=int(du)/(4-u^2)^(3/2)#
Now we will apply the trigonometric substitution #u=2sintheta#. Recall that this implies that #du=2costhetad theta#.
#I=int(2costhetad theta)/(4-4sin^2theta)^(3/2)#
#=int(2costhetad theta)/(4^(3/2)(1-sin^2theta)^(3/2))#
Note that #1-sin^2theta=cos^2theta#:
#I=int(2costhetad theta)/(8(cos^2theta)^(3/2))#
#=int(costhetad theta)/(4cos^3theta)#
#=1/4int(d theta)/cos^2theta#
#=1/4intsec^2thetad theta#
#=1/4tantheta+C#
Recall that #u=2sintheta#, so #theta=arcsin(u/2)#.
#I=1/4tan(arcsin(u/2))+C#
We can simplify this: draw the triangle where sine is #u/2#, that is, the opposite side is #u# and the hypotenuse is #2#. Through the Pythagorean theorem, we see that the adjacent side is #sqrt(4-u^2)#.

Thus, the tangent is opposite over adjacent, or:

#I=1/4(u/sqrt(4-u^2))+C#
Now, since #u=tanx#:
#I=tanx/(4sqrt(4-tan^2x))+C#
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Answer 2

To integrate the given expression using trigonometric substitution, we can let ( \tan{x} = u ), so ( \sec^2{x} ) becomes ( 1 + u^2 ). Then, we substitute ( \sec^2{x} = 1 + u^2 ) and ( \tan^2{x} = u^2 ) into the expression. After substitution, we end up with ( \int \frac{1}{(4 - u^2)^{3/2}} , du ). Next, we can use a trigonometric identity to simplify this integral. By letting ( u = 2\sin{\theta} ), we can rewrite the expression in terms of ( \theta ). After making this substitution, we can integrate with respect to ( \theta ) and then convert back to the variable ( 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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