Algebra.Com's Answer #845132 by ikleyn(52794)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "f (x) is  polynomial function, f '(x) + int f (x) dx = x ^4 + 13 x ^2 + 2,
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document.write( "Find f(x)
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document.write( "We want to find f(x) as a polynomial f(x) =  +  + . . . +  .\r\n" );
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document.write( "Taking derivative decreases the degree of a polynomial by one unit.\r\n" );
document.write( "Taking antiderivative increases the degree of a polynomial by one unit.\r\n" );
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document.write( "Since the sum f ' (x) + int f (x) dx is a polynomial of degree 4,\r\n" );
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document.write( "it means that the sough polynomial f(x) is of degree 3:\r\n" );
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document.write( " f(x) = ax^3 + bx^2 + cx + d.\r\n" );
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document.write( "Then \r\n" );
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document.write( " f ' (x) =  + 2bx + c,\r\n" );
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document.write( " int f(x) dx =  +  +  + dx + E.\r\n" );
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document.write( "So, in the sum f ' (x) + int f(x) dx\r\n" );
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document.write( " (a) coefficient at  is  It gives an equation\r\n" );
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document.write( " = 1; hence a = 4.\r\n" );
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document.write( " (b) coefficient at  is 0. It gives an equation\r\n" );
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document.write( "  = 0; hence b = 0.\r\n" );
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document.write( " (c) coefficient at  is 13. It gives an equation\r\n" );
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document.write( "  = 13, or  = 13 --->  = 13 - 12 = 1 ---> c = 2.\r\n" );
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document.write( " (d) coefficient at  is 0. It gives an equation\r\n" );
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document.write( " 2b + d = 0, which implies 2*0 + d = 0; hence, d = 0.\r\n" );
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document.write( " +------------------------------------------------------------+\r\n" );
document.write( " | At this point, the problem is just solved to the end. |\r\n" );
document.write( " | a = 4; b = 0; c = 2; d = 0. |\r\n" );
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document.write( "The sough polynomial is f(x) = 4x^3 + 2x. ANSWER\r\n" );
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document.write( "CHECK. The derivative is f ' (x) =  .\r\n" );
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document.write( " The anti-derivative is F(x) =  =  .\r\n" );
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document.write( " The sum f ' (x) + F(x) =  +  =  . ! correct !\r\n" );
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document.write( "Solved.\r
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document.write( "Do not accept any other  answer.\r
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document.write( "The solution by Edwin is INCORRECT.\r
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document.write( "To make sure that it is incorrect, simply take the antiderivative of his leading term  .\r
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document.write( "This antiderivative is  =  , and no other arguments are needed anymore.\r
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