Question

integretionx^4+1/x^2+1 dx​

Answer 1

$( {x}^{5} \div 5) - (1 \div x) + x$

Answer 2

$\underline{\textsf{Solution :}}$

$\mathsf{Now,\:\frac{x^{4}+1}{x^{2}+1}}$

$\mathsf{=\frac{(x^{2}+1)^{2}-2x^{2}}{x^{2}+1}}$

$\mathsf{=\frac{(x^{2}+1)^{2}}{x^{2}+1}-\frac{2x^{2}}{x^{2}+1}}$

$\mathsf{=x^{2}+1-2\frac{(x^{2}+1)-1}{x^{2}+1}}$

$\mathsf{=x^{2}+1-2+\frac{2}{x^{2}+1}}$

$\mathsf{=x^{2}-1+\frac{2}{x^{2}+1}}$

$\to \mathsf{\frac{x^{4}+1}{x^{2}+1}=x^{2}-1+\frac{2}{x^{2}+1}}$

$\textsf{On integration, we get}$

$\mathsf{\int \frac{x^{4}+1}{x^{2}+1}=\int x^{2}dx-\int dx+\int\frac{2}{x^{2}+1}}$

$\mathsf{=\frac{x^{3}}{3}-x+2\:tan^{-1}x+C}$

$\textsf{where C is integral constant}$

$\to \boxed{\small{\mathsf{\int \frac{x^{4}+1}{x^{2}+1}=\frac{x^{3}}{3}-x+2\:tan^{-1}x+C}}}$