document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=819248>Question 819248</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the most general antiderivative of each function.<BR>\n" );
document.write( "a.) f(x)=20x^3+12x^2+4<BR>\n" );
document.write( "b.) g(x)=3x^4-x^3+6x^2/x^4<BR>\n" );
document.write( "c.) h(t)=e^x+sinx+14(5root(x^2)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #493027 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=493027\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The phrase \"most general\" must mean that you need to add a constant.<BR>\n" );
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document.write( "a.) The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E4%2B4x%5E3%2B4x ALIGN=MIDDLE  ALT=\"5x%5E4%2B4x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=20x%5E3%2B12x%5E2%2B4 ALIGN=MIDDLE  ALT=\"f%28x%29=20x%5E3%2B12x%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285x%5E4%2B4x%5E3%2B4x%2BK%29 ALIGN=MIDDLE  ALT=\"highlight%285x%5E4%2B4x%5E3%2B4x%2BK%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K ALIGN=MIDDLE  ALT=\"K\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents any number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
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document.write( "b.) You wrote g(x)=3x^4-x^3+6x^2/x^4=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E4-x%5E3%2B6x%5E2%2Fx%5E4=3x%5E4-x%5E3%2B6x%5E%28-2%29 ALIGN=MIDDLE  ALT=\"3x%5E4-x%5E3%2B6x%5E2%2Fx%5E4=3x%5E4-x%5E3%2B6x%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The antiderivative of that would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E5%2F5-x%5E4%2F4%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x%5E5%2F5-x%5E4%2F4%2BK-6%2Fx ALIGN=MIDDLE  ALT=\"3x%5E5%2F5-x%5E4%2F4%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x%5E5%2F5-x%5E4%2F4%2BK-6%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "However, if you meant g'(x)=(3x^4-x^3+6x^2)/x^4=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E4-x%5E3%2B6x%5E2%29%2Fx%5E4=3-x%5E%28-1%29%2B6x%5E%28-2%29=3x-1%2Fx%2B6%2Fx%5E2 ALIGN=MIDDLE  ALT=\"%283x%5E4-x%5E3%2B6x%5E2%29%2Fx%5E4=3-x%5E%28-1%29%2B6x%5E%28-2%29=3x-1%2Fx%2B6%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the antiderivative is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-ln%28x%29%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x-ln%28x%29-6%2Fx%2BK ALIGN=MIDDLE  ALT=\"3x-ln%28x%29%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x-ln%28x%29-6%2Fx%2BK\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "c.) If you meant h(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex%2Bsin%28x%29%2B14root%285%2Cx%5E2%29=e%5Ex%2Bsin%28x%29%2B14x%5E%222%2F5%22 ALIGN=MIDDLE  ALT=\"e%5Ex%2Bsin%28x%29%2B14root%285%2Cx%5E2%29=e%5Ex%2Bsin%28x%29%2B14x%5E%222%2F5%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "The antiderivative of that function can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex%2Bsin%28x%29%2B14%28x%5E%28%222%2F5%22%2B1%29%2F%28%222%2F5%22%2B1%29%29=e%5Ex%2Bsin%28x%29%2B14x%5E%227%2F5%22%2F%287%2F5%29=e%5Ex%2Bsin%28x%29%2B14%285%2F7%29x%5E%227%2F5%22=e%5Ex%2Bsin%28x%29%2B10root%285%2Cx%5E7%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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