`int x tan^2 x dx` Evaluate the integral

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`intxtan^2xdx`

 Rewrite the integrand using the identity `tan^2x=sec^2x-1`

`intxtan^2xdx=intx(sec^2x-1)dx`

`=intxsec^2xdx-intxdx`

Now let's evaluate `intxsec^2xdx`  using integration by parts,

`intxsec^2xdx=x*intsec^2xdx-int(d/dx(x)intsec^2(x))dx`

`=xtan(x)-int(1*tan(x))dx`

`=xtan(x)-int(sin(x)/cos(x))dx`

Substitute cos(x)=t

-sin(x)dx=dt

`int(sin(x)/cos(x))dx=int-dt/t`

`=-ln|t|`

substitute back t=cos(x),

`=-ln|cos(x)|`

`intxsec^2xdx=xtan(x)-(-ln|cos(x)|)`

`=xtan(x)+ln|cos(x)|`

`intxtan^2(x)dx=xtan(x)+ln|cos(x)|-intxdx`

`=xtan(x)+ln|cos(x)|-x^2/2+C`

C is a constant

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial