What is the integral of #int tan^4x dx#?

Question

Samantha Adkison · Accepted Answer

#(tan^3x)/3-tanx+x+C#

Solving trig antiderivatives usually involves breaking the integral down to apply Pythagorean Identities, and them using a #u#-substitution. That's exactly what we'll do here.

Begin by rewriting #inttan^4xdx# as #inttan^2xtan^2xdx#. Now we can apply the Pythagorean Identity #tan^2x+1=sec^2x#, or #tan^2x=sec^2x-1#: #inttan^2xtan^2xdx=int(sec^2x-1)tan^2xdx# Distributing the #tan^2x#: #color(white)(XX)=intsec^2xtan^2x-tan^2xdx# Applying the sum rule: #color(white)(XX)=intsec^2xtan^2xdx-inttan^2xdx#

We'll evaluate these integrals one by one.

First Integral This one is solved using a #u#-substitution: Let #u=tanx# #(du)/dx=sec^2x# #du=sec^2xdx# Applying the substitution, #color(white)(XX)intsec^2xtan^2xdx=intu^2du# #color(white)(XX)=u^3/3+C# Because #u=tanx#, #intsec^2xtan^2xdx=(tan^3x)/3+C#

Second Integral Since we don't really know what #inttan^2xdx# is by just looking at it, try applying the #tan^2=sec^2x-1# identity again: #inttan^2xdx=int(sec^2x-1)dx# Using the sum rule, the integral boils down to: #intsec^2xdx-int1dx# The first of these, #intsec^2xdx#, is just #tanx+C#. The second one, the so-called "perfect integral", is simply #x+C#. Putting it all together, we can say: #inttan^2xdx=tanx+C-x+C# And because #C+C# is just another arbitrary constant, we can combine it into a general constant #C#: #inttan^2xdx=tanx-x+C#

Combining the two results, we have: #inttan^4xdx=intsec^2xtan^2xdx-inttan^2xdx=((tan^3x)/3+C)-(tanx-x+C)=(tan^3x)/3-tanx+x+C#

Again, because #C+C# is a constant, we can join them into one #C#.

Scarlett Allison · Answer

undefined The integral of $ \int \tan^4(x) \, dx $ can be evaluated using trigonometric identities and integration techniques. One approach is to express $ \tan^4(x) $ in terms of $ \sin(x) $ and $ \cos(x) $ using the identity $ \tan^2(x) = \sec^2(x) - 1 $. Then, you can integrate term by term. After integrating, you'll get the result in terms of trigonometric functions and constants.