The correct option is B 1/(x^2+1) 7/(x^2+1)^2+7/(x^2+1)^3 Let us use substitution method.Let x^2+1=k⇒ x^2=k 1 substi in x^4 5x^2+1(x^2+1)^3 we g ...

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Integration by Partial Fractions

x4-5x2+1x2+13...

Question

$\frac{x^{4} - 5 x^{2} + 1}{(x^{2} + 1)^{3}} =$

\frac{1}{(x^{2} + 1)} - \frac{1}{(x^{2} + 1)^{2}} + \frac{7}{(x^{2} + 1)^{3}}

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\frac{1}{(x^{2} + 1)} - \frac{7}{(x^{2} + 1)^{2}} + \frac{7}{(x^{2} + 1)^{3}}

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\frac{7}{(x^{2} + 1)} - \frac{7}{(x^{2} + 1)^{2}} + \frac{1}{(x^{2} + 1)^{3}}

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\frac{7}{(x^{2} + 1)} - \frac{1}{(x^{2} + 1)^{2}} - \frac{1}{(x^{2} + 1)^{3}}

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Solution

The correct option is B $\frac{1}{(x^{2} + 1)} - \frac{7}{(x^{2} + 1)^{2}} + \frac{7}{(x^{2} + 1)^{3}}$
Let us use substitution method.
Let $x^{2} + 1 = k \Rightarrow x^{2} = k - 1$
substi in $\frac{x^{4} - 5 x^{2} + 1}{(x^{2} + 1)^{3}}$
we get $\frac{k^{2} + 1 - 2 k - 5 (k - 1) + 1}{k^{3}} = \frac{k^{2} - 7 k + 7}{k^{3}}$
$\Rightarrow 1 / k - 7 / k^{2} + 7 / k^{3}$
re substitute the value of k
$= 1 / x^{2} + 1 - 7 / (x^{2} + 1)^{2} + 7 / (x^{2} + 1)^{3}$

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Similar questions

\frac{x^{4} + 24 x^{2} + 28}{(x^{2} + 1)^{3}}

= \frac{A x + B}{x^{2} + 1} + \frac{C x + D}{(x^{2} + 1)^{2}} + \frac{E x + F}{(x^{2} + 1)^{3}}

A =

\frac{3 x^{2} + 1}{(x^{2} + 1)^{3}} = \frac{A x + B}{(x^{2} + 1)} + \frac{C x + D}{(x^{2} + 1)^{2}} + \frac{E x + F}{(x^{2} + 1)^{3}}

A + C + E + F =

x^{2} - \frac{13}{24} x - \frac{1}{12}

{sin}^{- 1} (\frac{5 x + 12 \sqrt{1 - x^{2}}}{13})

Question :

One of the factors of x⁵+x+1 is

(1) 2x² - x - 1

(2) x³ - x²+ 1

(3) x² - 2x² + 1

(4) x³ + x² + 1

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