Maths-
General
Easy
Question
If
then ![P left parenthesis x right parenthesis equals](data:image/png;base64,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)
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 correct answer is: ![x plus a plus b plus c](data:image/png;base64,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)
Given :
![fraction numerator x to the power of 4 end exponent over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals P left parenthesis x right parenthesis plus fraction numerator A over denominator x minus a end fraction plus fraction numerator B over denominator x minus b end fraction plus fraction numerator C over denominator x minus c end fraction](data:image/png;base64,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)
To Find : P(x)
Let P(x) be a linear polynomial i.e. = mx + n ( since y = mx + c)
Substitute this value in the equation
![fraction numerator x to the power of 4 over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals m x space plus space n plus fraction numerator A over denominator x minus a end fraction plus fraction numerator B over denominator x minus b end fraction plus fraction numerator C over denominator x minus c end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaQAAAApCAYAAAB6FV2HAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAABrJJREFUeNrtnW9kXWccxx8RERUhqiquKTU1MxVmJqL2Ji8qqmLMxMxEiJmamTA1NVPlqqmZvamKmKlxTV9UzJiYiOqbicmLmVKzFzFTZmoiItw935zfldPT85x7zrnn3vP8+X74kfvcG/f3Pfc8v+f/7yhFquIdbW1eBkIIIXXyirYHbJAIIYTUyYS2h9oabJAIIYTUyffapuVvNkiEEEJq4RNt78Zes0EiofFRoLr3tR1q+1PbY233tU0Gov2stpvaftW2J9fguopmi0iNtA1GSAi8re1A23Bgul+UYBwHAXktAO0r2ta1XdA2JGUvSMdkjVXCvgaKkBBAb3hH20/a5gPTDr13E2VvppT5BmaE7vDWHzxL2pop5dflPTZIJHRua1vQ9p62rwPT/pkE5zjfaLvoseZzKpqaHOatXw/b2k7HXi/mqHhskEgIYBPPuvx9WgJVSLRklIQpqyltq8r/tbQvVLjrhVaA3s4t+XtW2wYvCSFHPeRfVLRuEO+8vRTQNXikjteL//N8ZNThNxWtnZEawfz4GypawDzJy0HI0XTVSqLshrYPAtE/JI0QGJHOKs4hnvNc8z5v/frBEBW7iM7zUhByNAraTilHp+1+INfgdfX8bAnSht30WDMa3qe8/etlRkWHXjFtt8DLQYjaUuajDqFs/0YsWE2U4Syi77vPMCo8wSpQD2elx4cfYExFc+acsiMhs6zty4z3v9M2F8B1+EpFG5zirAbQaYXGFVaDwXNK2w+JBuiStm95aUigIAPBjnTOTCx1abB84Z463sSAWPGWitaYRzzXfUbbP9ququOMDKMSG9FYzbKaVM+I3HANQw/wIi8RCZCWBJ4sUGceBXAtEJQ705T7EhcagdwH2GWH81ZPRf+/KlrWmGMVIYQQQgghR7QD/35qp35CgqWTsgXZAbC4/Le2v9TxlAHmtrErEtMqmFa4Gvvfsumg+hUUOud5oAVJKffE7w8ZlKid+gkZXEXqZiZaEsix6xG7mrDNGIutf6ho4X1TRRmoUY5MAvvq2fRPZdJB9SsotKTxQWOEczw4fNgQn08M4PsZkKmfENIDWDTfUM8/d2VXRedA0srjz6ipOh1Uu0ctyynlGCWdHMD3MyBTv/M9Vxotr1UNRhB7KaOHbuVJekkHVZXuePqZOMMGn6v4ftfvmxC0h6yfEKem7JCy5ecKyqtMB1W2Ipl8mzaUc5RA7dRPiEVgzWitx/Kq00G1B6SFQYnaqZ8Qi8CuuuUeyvuRDqrdg5alAuUMStRO/SpKKd+02L+myk57b7v/rmvo5nuVIIPGXMnyfqWDag9IC4MStQevH3PsDx3w84FKXw9wxX/XNZh8rxrsPhstUW5jOqiiWggJHgSaKQf8fFVFafFt9R+7poYc11DGd0JIsThADExJMHSFLQmMtvmPZIQ7jmso4zshpFwcICng6YdXHPIXJ96bFvo/r6JpIZc1lPGdEFIuDpAUftR2IaW8H/nAssAQ93MV5SzD2REcZjyT8jmc39iw0H+kuflUNCDnGtLC4JzJpEMawJg0kLvyO2BNZiLDd2I/ttZlH7UXiQM+6s8TQzJBYkrTA6eqzgeWxbqKtsJOyA3dkt5GkuQz323xH/4+VlFC0I6GW4bekq0axuT7bmgbF5+QFPRShu/EDWysyz5qLxIHfNSfJ4ZkcpjxXtX5wEwsyA8ZBw+HMm2LPbDMf4B8ZS8nysYzgreNGpoSSLpxwPjuHLbWZZ+0l4kDvunPG0NKNUigSD6wsqliMKSdTpRtZgzzDyzz35RLbbSCBmmQGp7kHFqzQXITG+uyT9rLxAHf9OeNIUaypotAlfnATCS3SZoSUqZNGdngf1a+sg1HNLym8m3p5pSdu9hWl33TXjQO+KY/bwzp2nLOGN6rOh+YiScp37tt+Gzyx7XBf1Neso+1XXNEA6ZU7uX4HDc1uImNddk37UXjgG/688aQTExbjvuRD8zErnr21Po1uYBpXBGfbfIfc6aLKSMJnEeYdEQDzhb9nuNzSd+J/dhal33TXjQO+KY/bwzJJO1QZr/ygZnAFsTbMrzHXPNd+f408hwqHbT/6BVgMXNWXjekbNEhDQDzy9gdg2f1jMswf6aL78RubK7LvmkvGgd80583hnQFOdTOx1r0OvKBYe9+S3pU6GHhDMN8Sgu8ZaH/8PWy3IyH8qPMZ/Qitiz9DfAo8E3RgF7dUk7fiZ3YXJd91F4kDvj623eLIbmwMbHnqZQyJle103dCitZlQjJ5X9mdEgbrFssO+++6hm6+E0JIbv4H/siR1+hbRhQAAAJMdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtc3VwPjxtaT54PC9taT48bW4+NDwvbW4+PC9tc3VwPjxtcm93Pjxtbz4oPC9tbz48bWk+eDwvbWk+PG1vPi08L21vPjxtaT5hPC9taT48bW8+KTwvbW8+PG1vPig8L21vPjxtaT54PC9taT48bW8+LTwvbW8+PG1pPmI8L21pPjxtbz4pPC9tbz48bW8+KDwvbW8+PG1pPng8L21pPjxtbz4tPC9tbz48bWk+YzwvbWk+PG1vPik8L21vPjwvbXJvdz48L21mcmFjPjxtbz49PC9tbz48bWk+bTwvbWk+PG1pPng8L21pPjxtbz4mI3hBMDs8L21vPjxtbz4rPC9tbz48bW8+JiN4QTA7PC9tbz48bWk+bjwvbWk+PG1vPis8L21vPjxtZnJhYz48bWk+QTwvbWk+PG1yb3c+PG1pPng8L21pPjxtbz4tPC9tbz48bWk+YTwvbWk+PC9tcm93PjwvbWZyYWM+PG1vPis8L21vPjxtZnJhYz48bWk+QjwvbWk+PG1yb3c+PG1pPng8L21pPjxtbz4tPC9tbz48bWk+YjwvbWk+PC9tcm93PjwvbWZyYWM+PG1vPis8L21vPjxtZnJhYz48bWk+QzwvbWk+PG1yb3c+PG1pPng8L21pPjxtbz4tPC9tbz48bWk+YzwvbWk+PC9tcm93PjwvbWZyYWM+PC9tYXRoPuj5mjoAAAAASUVORK5CYII=)
Taking LCM
![fraction numerator x to the power of 4 over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals fraction numerator left parenthesis m x space plus space n right parenthesis left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis space plus space A left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis plus space B left parenthesis x minus a right parenthesis left parenthesis x minus c right parenthesis space plus space C left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis space space over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction](data:image/png;base64,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)
Cancelling denominators on both sides
![x to the power of 4 over blank equals fraction numerator left parenthesis m x space plus space n right parenthesis left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis space plus space A left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis plus space B left parenthesis x minus a right parenthesis left parenthesis x minus c right parenthesis space plus space C left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis space space over denominator blank end fraction](data:image/png;base64,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)
On comparing the coefficients of
from both sides ,we get
m = 1
-am -bm -cm + n = 0
a + b + c = n
P(x) =mx + n
Substituting the values of m and n in
P(x) = x + a + b + c
Related Questions to study
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
maths-
Statement I : y = f(x) =
, x
R Range of f(x) is [3/4, 1)
Statement II :
.
Statement I : y = f(x) =
, x
R Range of f(x) is [3/4, 1)
Statement II :
.
maths-General
maths-
Statement I : Function f(x) = sinx + {x} is periodic with period ![2 pi](data:image/png;base64,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)
Statement II : sin x and {x} are both periodic with period
and 1 respectively.
Statement I : Function f(x) = sinx + {x} is periodic with period ![2 pi](data:image/png;base64,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)
Statement II : sin x and {x} are both periodic with period
and 1 respectively.
maths-General