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Question

The partial fractions of $fraction numerator x to the power of 2 end exponent over denominator open parentheses x to the power of 2 end exponent plus a to the power of 2 end exponent close parentheses open parentheses x to the power of 2 end exponent plus b to the power of 2 end exponent close parentheses end fraction$ are

$\frac{1}{a^{2} + b^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$
$\frac{1}{b^{2} - a^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$
$\frac{1}{a^{2} - b^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$
$\frac{1}{a^{2} - b^{2}} [\frac{1}{x^{2} + a^{2}} - \frac{1}{x^{2} + b^{2}}]$

hint Hint:

In order to integrate a rational function, it is reduced to a proper rational function. The method in which the integrand is expressed as the sum of simpler rational functions is known as decomposition into partial fractions. After splitting the integrand into partial fractions, it is integrated accordingly with the help of traditional integrating techniques.
The stepwise procedure for finding the partial fraction decomposition is explained here::

Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.

Step 2: Now, factor the denominator of the rational expression into the linear factor or in the form of irreducible quadratic factors (Note: Don’t factor the denominators into the complex numbers).

Step 3: Write down the partial fraction for each factor obtained, with the variables in the numerators, say A and B.

Step 4: To find the variable values of A and B, multiply the whole equation by the denominator.

Step 5: Solve for the variables by substituting zero in the factor variable.

Step 6: Finally, substitute the values of A and B in the partial fractions.

The correct answer is: $fraction numerator 1 over denominator a to the power of 2 end exponent minus b to the power of 2 end exponent end fraction open square brackets fraction numerator a to the power of 2 end exponent over denominator x to the power of 2 end exponent plus a to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator x to the power of 2 end exponent plus b to the power of 2 end exponent end fraction close square brackets$

Given :
$fraction numerator x to the power of 2 end exponent over denominator open parentheses x to the power of 2 end exponent plus a to the power of 2 end exponent close parentheses open parentheses x to the power of 2 end exponent plus b to the power of 2 end exponent close parentheses end fraction$

Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
Let $x squared space equals space z$
$fraction numerator z over denominator open parentheses z plus a squared close parentheses open parentheses z plus b squared close parentheses end fraction$

Step 2: Write down the partial fraction for each factor obtained, with the variables in the numerators, say A and B.
$fraction numerator z over denominator open parentheses z plus a squared close parentheses open parentheses z plus b squared close parentheses end fraction space equals space fraction numerator A over denominator z plus a squared end fraction space plus space fraction numerator B over denominator z plus b squared end fraction$

Step 3: To find the variable values of A and B, multiply the whole equation by the denominator.
$fraction numerator z over denominator open parentheses z plus a squared close parentheses open parentheses z plus b squared close parentheses end fraction space equals space fraction numerator A over denominator z plus a squared end fraction space plus space fraction numerator B over denominator z plus b squared end fraction$

On LHS take LCM and cancel the denominators on both side
. $rightwards double arrow z over blank space equals space fraction numerator A left parenthesis z plus b squared right parenthesis over denominator blank end fraction space plus space fraction numerator B left parenthesis z plus a squared right parenthesis over denominator blank end fraction rightwards double arrow z space equals space A z plus A b squared space space plus space B z plus B a squared S I m p l i f y i n g rightwards double arrow z space equals z space left parenthesis A plus space B right parenthesis space plus space A b squared space space plus B a squared C o m p a r i n g space L H S thin space a n d space R H S rightwards double arrow A space plus space B space equals space 1 space a n d space space A b squared space space plus B a squared space equals space 0 space rightwards double arrow space space A b squared space space equals space minus B a squared space rightwards double arrow space A space equals space minus fraction numerator B a squared over denominator b squared end fraction space S u b s t i t u t e space t h e space v a l u e space o f space A space i n rightwards double arrow A space plus space B space equals space 1 rightwards double arrow negative fraction numerator B a squared over denominator b squared end fraction space plus space B space equals space 1 rightwards double arrow B left parenthesis 1 minus space a squared over b squared right parenthesis space equals space 1 rightwards double arrow B space equals space fraction numerator b squared over denominator b squared space minus space a squared end fraction S u b s t i t u t e space t h e space v a l u e space o f space B space space i n space space A space equals space minus fraction numerator B a squared over denominator b squared end fraction space rightwards double arrow A space equals space minus fraction numerator b squared cross times a squared over denominator b squared left parenthesis b squared space minus space a squared right parenthesis end fraction space space equals space fraction numerator a squared over denominator b squared space minus space a squared end fraction$

Step 4: Finally, substitute the values of A and B in the partial fractions

$fraction numerator x to the power of 2 end exponent over denominator open parentheses x to the power of 2 end exponent plus a to the power of 2 end exponent close parentheses open parentheses x to the power of 2 end exponent plus b to the power of 2 end exponent close parentheses end fraction$ = $fraction numerator 1 over denominator a to the power of 2 end exponent minus b to the power of 2 end exponent end fraction open square brackets fraction numerator a to the power of 2 end exponent over denominator x to the power of 2 end exponent plus a to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator x to the power of 2 end exponent plus b to the power of 2 end exponent end fraction close square brackets$

Related Questions to study

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parallel

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parallel

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parallel

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parallel

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parallel

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