Mathematics

Solve the equation:-
$$\displaystyle \int \dfrac{x^{6}+1}{x^{2}+1}\ dx$$


SOLUTION
Solve the equation 
$$\displaystyle\int{\dfrac{x^{6}+1}{x^{2}+1}}dx$$
When degree of polynomial in numerator is higher then polynomial than first divide both the polynomial 
$$\qquad \qquad { x }^{ 4 }-{ x }^{ 2 }+1\\ \left. { x }^{ 2 }+1 \right) \overline { { x }^{ 6 }\quad \quad \quad \quad \quad \quad +1 } \\ \qquad \qquad { x }^{ 6 }+{ x }^{ 4 }\\ \qquad \qquad \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \qquad \qquad -{ x }^{ 4 }\quad \quad \quad \quad +1\\ \qquad \qquad -{ x }^{ 4 }-{ x }^{ 2 }\\ \qquad \qquad \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \qquad \qquad \qquad { x }^{ 2 }+1\\ \qquad \qquad \qquad { x }^{ 2 }+1\\ \qquad \qquad \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ $$
$$=\displaystyle\int\left(x^{4}-x^{2}+1\right)dx$$
$$=\displaystyle\int x^{4}dx-\int x^{2}dx +\int dx$$
$$=\dfrac{x^{5}}{5}-\dfrac{x^{3}}{3}+x+c$$

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Subjective Medium Published on 17th 09, 2020
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