Question

$\int\limits^2_1 x²+1/x⁴+1 dx ,Evaluate the given integral expression:$

Answer 1

HELLO DEAR,



GIVEN:-
$\sf{\int\limits^2_1 (x^2+1)/(x^4+1) dx}$


= $\sf{\int\limits^2_1 {(1 + 1/x^2)\over(x^2 + 1/x^2)}\,dx}$ [ dividing num.and denom. by x²]

= $\sf{\int\limits^2_1 {(1 + 1/x^2)\over{(x - 1/x)^2 + 2}}\,dx}$

= $\sf{\int\limits^{3/2}_0 dt/[t^2 + (\sqrt{2})^2]}$
where,(x - 1/x) = t and (1 + 1/x²).dx = dt
Also, [x = 2 , t = 3/2] and [x = 1 , t = 0]

= $\sf{1/(\sqrt{2})[tan^{-1}(t/\sqrt{2})]^{^{3/2}}_0}$

= $\sf{1/(\sqrt{2})[tan^{-1}(3/2\sqrt{2}) - tan^{-1}0]}$

= $\sf{1/(\sqrt{2})[tan^{-1}(3/2\sqrt{2})]}$



I HOPE ITS HELP YOU DEAR,
THANKS