Question #6f0de

2 Answers
Nov 22, 2017

x2x6+9dx=π9

Explanation:

x2x6+9dx

=133x2dx(x3)2+9

After using y=x3 an dy=3x2dx transformation, this integral became

13dyy2+9

=230dyy2+9

=29[arctan(y3)]0

=29[arctan()arctan(0)]

=29(π20)

=π9

Nov 22, 2017

2nd way: I didn't use symmetry

x2x6+9dx

=133x2dx(x3)2+9

After using y=x3 an dy=3x2dx transformation, this integral became

13dyy2+9

Now I divided range of integral. First of it from to a and second of it from a to .

13adyy2+9+13adyy2+9

=19[arctan(y3)]a+19[arctan(y3)]a

=19[arctan(a3)arctan()]+19[arctan()arctan(a3)]

=19[arctan()arctan()]

=19[π2(π2)]

=π9