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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$