Maths-
General
Easy
Question
The partial fractions of
are
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](data:image/png;base64,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)
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](data:image/png;base64,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)
Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
Let ![x squared space equals space z](data:image/png;base64,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)
![fraction numerator z over denominator open parentheses z plus a squared close parentheses open parentheses z plus b squared close parentheses end fraction](data:image/png;base64,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)
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](data:image/png;base64,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)
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](data:image/png;base64,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)
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](data:image/png;base64,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)
Step 4: Finally, substitute the values of A and B in the partial fractions
= ![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](data:image/png;base64,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)
Related Questions to study
Maths-
The partial fractions of
are
The partial fractions of
are
Maths-General
Maths-
If
then ![P left parenthesis x right parenthesis equals](data:image/png;base64,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)
If
then ![P left parenthesis x right parenthesis equals](data:image/png;base64,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)
Maths-General
Maths-
Let a,b,c such that
then ,
?
Let a,b,c such that
then ,
?
Maths-General
Maths-
If
then B=
If
then B=
Maths-General
Maths-
If the remainders of the polynomial f(x) when divided by
are 2,5 then the remainder of
when divided by
is
If the remainders of the polynomial f(x) when divided by
are 2,5 then the remainder of
when divided by
is
Maths-General
Maths-
Assertion :
can never become positive.
Reason : f(x) = sgn x is always a positive function.
Assertion :
can never become positive.
Reason : f(x) = sgn x is always a positive function.
Maths-General
Maths-
=
=
Maths-General
maths-
Evaluate
dx
Evaluate
dx
maths-General
maths-
If
. The integral of
with respect to
is -
If
. The integral of
with respect to
is -
maths-General
maths-
is
is
maths-General
maths-
dx equals:
dx equals:
maths-General
maths-
If I =
, then I equals:
If I =
, then I equals:
maths-General
maths-
The indefinite integral of
is, for any arbitrary constant -
The indefinite integral of
is, for any arbitrary constant -
maths-General
maths-
If f
= x + 2 then
is equal to
If f
= x + 2 then
is equal to
maths-General
maths-
Let
, then
is equal to:
Let
, then
is equal to:
maths-General