What is the integral of #int tan^3(x) dx#?

Answer 1

#tan^2(x)/2+ln(abscos(x))+C#

Split up #tan^3(x)# into #tan^2(x)tan(x)# then rewrite #tan^2(x)# using the identity #tan^2(theta)+1=sec^2(theta)=>tan^2(theta)=sec^2(theta)-1#.
#inttan^3(x)dx=inttan^2(x)tan(x)dx=int(sec^2(x)-1)tan(x)dx#

Distribute:

#=intsec^2(x)tan(x)dx-inttan(x)dx#
For the first integral, apply the substitution #u=tan(x)=>du=sec^2(x)dx#, both of which are already in the integral.
#=intucolor(white).du-inttan(x)dx#
#=u^2/2-inttan(x)dx#
#=tan^2(x)/2-inttan(x)dx#
Now rewrite #tan(x)# as #sin(x)/cos(x)# and apply the substitution #v=cos(x)=>dv=-sin(x)dx#.
#=tan^2(x)/2-intsin(x)/cos(x)dx#
#=tan^2(x)/2+int(-sin(x))/cos(x)dx#
#=tan^2(x)/2+int(dv)/v#

This is a common integral:

#=tan^2(x)/2+ln(absv)+C#
#=tan^2(x)/2+ln(abscos(x))+C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The integral of ( \int \tan^3(x) , dx ) can be evaluated using the substitution method. Let ( u = \tan(x) ), then ( du = \sec^2(x) , dx ). This implies ( dx = \frac{du}{\sec^2(x)} = \cos^2(x) , du ).

So the integral becomes:

[ \int \tan^3(x) , dx = \int u^3 \cos^2(x) , du ]

Now, we need to express ( \cos^2(x) ) in terms of ( u ). Since ( u = \tan(x) ), we have ( \tan^2(x) + 1 = \sec^2(x) ). This implies ( 1 + u^2 = \sec^2(x) ), so ( \cos^2(x) = \frac{1}{1 + u^2} ).

Substituting this into the integral:

[ \int \tan^3(x) , dx = \int u^3 \cdot \frac{1}{1 + u^2} , du ]

This is a rational function and can be integrated by techniques like partial fraction decomposition. After integration, you'll get the final result.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7