How do you integrate # sec^3x (tanx) dx#?

Answer 1

#sec^3x/3+C#

When working with integrals of secant and tangent, it's important to remember the following:

Here, we see that we can write #sec^3x(tanx)# as #sec^2x(secxtanx)#, which is perfect, since it composed of #sec^2x# and the derivative of secant, #secxtanx#. This indicates to us that we want to use a substitution of #u=secx#.
#intsec^3x(tanx)dx=intsec^2x(secxtanx)dx#
With #u=secx# and #du=(secxtanx)dx#:
#=intu^2du=u^3/3+C=sec^3x/3+C#
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Answer 2

To integrate sec^3(x) tan(x) dx, you can use a substitution method. Let u = sec(x) + tan(x). Then, du = (sec(x)tan(x) + sec^2(x)) dx. Now, integrate u^3 du and then substitute back for u. The result will be (1/2) sec(x)tan(x) + (1/2) ln|sec(x) + tan(x)| + C, where C is the constant of integration.

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Answer 3

To integrate ( \sec^3(x) \tan(x) ) with respect to ( x ), we can use a u-substitution. Let ( u = \sec(x) ) and ( dv = \sec^2(x) \tan(x) , dx ). Then, ( du = \sec(x) \tan(x) , dx ) and ( v = \tan(x) ).

Applying integration by parts:

[ \int \sec^3(x) \tan(x) , dx = \sec(x) \tan(x) - \int \tan^2(x) \sec(x) , dx ]

Now, we can use the trigonometric identity ( \tan^2(x) = \sec^2(x) - 1 ) and substitute:

[ \int \sec^3(x) \tan(x) , dx = \sec(x) \tan(x) - \int (\sec^2(x) - 1) \sec(x) , dx ]

[ = \sec(x) \tan(x) - \int \sec^3(x) , dx + \int \sec(x) , dx ]

[ = \sec(x) \tan(x) - \int \sec(x) , dx + \int \sec(x) , dx ]

[ = \sec(x) \tan(x) + \ln|\sec(x) + \tan(x)| + C ]

So, the integral of ( \sec^3(x) \tan(x) ) with respect to ( x ) is ( \sec(x) \tan(x) + \ln|\sec(x) + \tan(x)| + C ), where ( C ) is the constant of integration.

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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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