Question

give me its answer only .. please ......
i will mark you brilliant​

Answer 1

Given :

• $I = ∫ \frac{ x^2 + 1 }{ x^4 - x^2 + 1 } dx$

To find :

• $I = ∫ \frac{ x^2 + 1 }{ x^4 - x^2 + 1 } dx$

According to the question :

⟹ $I = ∫ \frac{ x^2 + 1 }{ x^4 - x^2 + 1 } dx$

⟹ $∫ \frac{ x^2 + 1}{x^2 ( x^2 - 1 ) + ( \frac{1}{x^2})} dx$

⟹ $∫ \frac{1 + ( \frac{1}{x^2} )}{x^2 - 2 + ( \frac{1}{x^2} + 1 )} dx$

⟹ $∫ \frac{ 1 + ( \frac{1}{x^2} )}{x - ( \frac{1}{x})^2 + 1^2 } dx$

Put :

➳ $t = x - ( \frac{1}{x} )$

➳ $dt = ( 1 + ( \frac{1}{x^2} ) dx$

➳ $I = ∫ ( \frac{dt}{t^2 + 1^2}) dx$

We get :

⇢$I = tan^-1 ( t ) + C$

⇢$I = tan^-1 ( x - ( \frac{1}{x} ) + C$

⇢$\bold{ I = tan^-1 ( \frac{x^2 - 1 }{ x } ) + C }$

So, It's Done !!