i need help find the intergral for?

Question

f(x) = (2+x^2) /( 1 +x^2)

Answer 1

Perhaps easiest:

(2 + x²)/(1 + x²) = (1 + 1 + x²)/(1 + x²) = 1/(1 + x²) + 1

Each of these is readily integrated.

Answer 2

Seiche has used one of the shorter way, (which i personally prefer as am a lazy one) but if you wish to go the long way, here it is

Use Integration by Trigonometric Substitution

So (2+x^2) / ( 1 +x^2) = 2 / ( 1 + x^2) + x^2/( 1 + x^2)

Let solve the first term first

The first term

integral of 2 / ( 1 + x^2)

Let x = a * tan(y) , where a is the constant added to x^2 at the denominator in our case a = 1; for some angle y
so
x = a * tan( y) = tan(y), therefore

dx = sec^2(y) dy

y = arctan(x)

Take the denominator ( 1 + x^2 ) = 1 + tan^2(y) = sec^2(y)

so

2 / ( 1 + x^2) is

integral [ 2 * sec^2(y) dy /( sec^2(y) ) ] =

integral [ 2 dy]

--------------------------
= 2 arctan(x)
------------------------


so for the second term, x^2/( 1 + x^2)
using the same as above

x = a * tan( y) = tan(y),

dx = sec^2(y) dy

y = arctan(x)

then integral [ x^2/( 1 + x^2) ] =

integral [tan^2(y) * sec^2(y)dy / sec^2(y)dy] =

integral [ tan^2(y) dy ] = tan (y) - y : remember y = arctan(x)


= tan(arctan(x) ) - arctan(x) =
----------------------
x - arctan(x)
--------------------

so add

2 arctan(x) + x - arctan(x)

your answer is

x + arctan(x)

hope it helps

Answer 3

=x+arctan x
details: (2+x^2)(1+x^2)=1/(1+x^2)+1
integral----> x+atanx