IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

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the value of integration (dx/(a2+x2)7) between d limits 0 and infinity is equal to____???

mathwiz

this question's soln can be found here...
http://www.goiit.com/posts/list/integration-plz-solve-53704.htm

edison

Q) the value of integration (dx/(a2+x2)7) between d limits 0 and infinity is equal to____???

Sol) Substitute x=a tan

Thus dx/d

=a sec²

or dx = a sec²

d

...(1)

Also, a²+x²=a²(1+tan²

)=a²sec²

...(2)

When x = 0,

tends to 0; and when x = infinity,

tends to

/2

Thus, Using (1) and (2) the value of integral is

I = _{[ 0]}

^{[ /2]} a sec²

d

/(a²sec²

)⁷

or I =_{[ 0]}^{[ /2]} cos¹²

/a¹⁴

or I =1/a¹⁴_{[ 0]}^{[ /2]} cos¹²

d

...(3)

Now either use the reduction formula to solve this integral or use replace

by

/2 -

, and then add the so obtained value of I to (3) to obtain 2I and simplify.