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How to evaluate this problem x tan^2(x)dx from 0 to pi/3. I need help

How to evaluate this problem x tan^2(x)dx from 0 to pi/3. I need help

Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
Dear John
 

To calculate integral of xtan^2x.

We now that tan^2x = sec^2x-1.

Therefore Int xtan^2x dx = Int x*(sec^2x - 1)dx

Int xtan^2x dx= Int xsec2xdx - Int xdx

Int xtan^2x dx = Intx*sec^2xdx - x^2/2....(1)

Int xsec^2x dx = xtanx - Int{x'tanx}dx.

Int xtanx dx =  xtanx - Int tanx dx

Int xtanx = xtanx+log(cosx) + C. Substituting this in (1) we get:

Int x tan^2x dx = xtanx + log(cosx) - x^2/2+C.

 

Now put limit from 0 to pi/3

(Pi/3)* sqrt(3)+ log(1/2) - pi²/18

 

Regards

Arun (askIITians forum expert)

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