document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193654>Question 1193654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the solution to the integral of (ln^3 x) / (x)(√ln^2 x - 4) dx using trigonometric substitution? The answer should be equal to ((8+ln^2 x)/3)(√ln^2 x - 4). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848541 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848541\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's tackle this integral using trigonometric substitution.\r<BR>\n" );
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document.write( "**1. Preparation**\r<BR>\n" );
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document.write( "* **Substitution:** <BR>\n" );
document.write( "    * Let ln(x) = 2sec(θ) <BR>\n" );
document.write( "    * Then, d(ln(x)) = 2sec(θ)tan(θ) dθ <BR>\n" );
document.write( "    * Which implies: (1/x) dx = 2sec(θ)tan(θ) dθ\r<BR>\n" );
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document.write( "**2. Substitute into the Integral**\r<BR>\n" );
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document.write( "* The integral becomes: <BR>\n" );
document.write( "   ∫ (ln³(x)) / (x)(√(ln²(x) - 4)) dx <BR>\n" );
document.write( "   = ∫ (2sec(θ))³ * 2sec(θ)tan(θ) / (√(4sec²(θ) - 4)) dθ <BR>\n" );
document.write( "   = ∫ 16sec⁴(θ)tan(θ) / (2√(sec²(θ) - 1)) dθ <BR>\n" );
document.write( "   = ∫ 8sec⁴(θ)tan(θ) / tan(θ) dθ <BR>\n" );
document.write( "   = ∫ 8sec⁴(θ) dθ\r<BR>\n" );
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document.write( "**3. Simplify the Integral**\r<BR>\n" );
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document.write( "* Use the identity: sec²(θ) = 1 + tan²(θ)<BR>\n" );
document.write( "   ∫ 8sec⁴(θ) dθ = ∫ 8(1 + tan²(θ))sec²(θ) dθ <BR>\n" );
document.write( "   = ∫ 8sec²(θ) dθ + ∫ 8tan²(θ)sec²(θ) dθ\r<BR>\n" );
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document.write( "**4. Evaluate the Integrals**\r<BR>\n" );
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document.write( "* ∫ 8sec²(θ) dθ = 8tan(θ) + C₁ <BR>\n" );
document.write( "* ∫ 8tan²(θ)sec²(θ) dθ = 8/3 * tan³(θ) + C₂ \r<BR>\n" );
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document.write( "**5. Combine the Results**\r<BR>\n" );
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document.write( "* ∫ 8sec⁴(θ) dθ = 8tan(θ) + 8/3 * tan³(θ) + C \r<BR>\n" );
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document.write( "**6. Back-Substitute for x**\r<BR>\n" );
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document.write( "* Recall: ln(x) = 2sec(θ) <BR>\n" );
document.write( "   => sec(θ) = ln(x) / 2 <BR>\n" );
document.write( "   => tan(θ) = √(sec²(θ) - 1) = √((ln²(x) / 4) - 1) = √(ln²(x) - 4) / 2\r<BR>\n" );
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document.write( "* Substitute these values back into the result:\r<BR>\n" );
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document.write( "   ∫ (ln³(x)) / (x)(√(ln²(x) - 4)) dx = 8 * (√(ln²(x) - 4) / 2) + 8/3 * (√(ln²(x) - 4) / 2)³ + C <BR>\n" );
document.write( "   = 4√(ln²(x) - 4) + (1/3) * √(ln²(x) - 4)³ + C\r<BR>\n" );
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document.write( "**Therefore, the solution to the integral is:**\r<BR>\n" );
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document.write( "∫ (ln³(x)) / (x)(√(ln²(x) - 4)) dx = 4√(ln²(x) - 4) + (1/3) * √(ln²(x) - 4)³ + C \r<BR>\n" );
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document.write( "I hope this comprehensive solution is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
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