document.write( "Question 1208301: Hi can you help please:
\n" ); document.write( "int (3 x^3) (sqrt(16-x^2)) dx
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Algebra.Com's Answer #846634 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Here is the way to solve it by trig substitution. It isn't so bad after all.\r\n" );
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document.write( "\"int%28%283x%5E3%29+%28sqrt%2816-x%5E2%29%29%2Cdx%29\"\r\n" );
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document.write( "Draw a right tringle with angle θ, hypotenuse 4, opposite leg x, and\r\n" );
document.write( "adjacent leg \"sqrt%2816-x%5E2%29\". As a convention we always put x on the opposite\r\n" );
document.write( "leg [although it wouldn't matter if we used cofunctions instead.]\r\n" );
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document.write( "[I can't write dθ using the program on the site, so I'll use \"dO\" for \"dθ\", so don't be upset.\r\n" );
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document.write( "\"int%28%283x%5E3%29+%28sqrt%2816-x%5E2%29%29%2Cdx%29\"\r\n" );
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document.write( "Use the substitutions above, taken straight from the right triangle:\r\n" );
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document.write( "\"int%28%283%284sin%28theta%29%29%5E3%29%284cos%28theta%29%29%2A4cos%28theta%29dO%29\"\r\n" );
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document.write( "\"int%28%283%2A4%5E3sin%5E3%28theta%29%5E%22%22%29%284cos%28theta%29%29%2A4cos%28theta%29dO%29\"\r\n" );
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document.write( "Take out all the constants:\r\n" );
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document.write( "\"3072int%28sin%5E3%28theta%29cos%5E2%28theta%29dO%29\"\r\n" );
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document.write( "When you have an odd power of a sine or cosine, save a sine or cosine\r\n" );
document.write( "for the derivative and replace the even power that's left with an\r\n" );
document.write( "identity. so we break up sin3θ = (sin2θ)sinθ = (1-cos2θ)sinθ\r\n" );
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document.write( "\"3072int%28%281-cos%5E2%28theta%29%29%2Asin%5E%22%22%28theta%29cos%5E2%28theta%29dO%29\"\r\n" );
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document.write( "\"3072int%28cos%5E2%28theta%29sin%28theta%29dO%29\"\"%22%22-%22%22\"\"3072int%28cos%5E4%28theta%29sin%28theta%29dO%29\"\r\n" );
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document.write( "We need to insert a - sign since the derivative of cosine is negative sine:\r\n" );
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document.write( " \"-3072int%28cos%5E2%28theta%29%28-sin%28theta%29%29dO%29\"\"%22%22%2B%22%22\"\"3072int%28cos%5E4%28theta%29%28-sin%28theta%29%29dO%29\"\r\n" );
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document.write( "\"-3072%28cos%5E3%28theta%29%2F3%5E%22%22%29%2B3072%28cos%5E5%28theta%29%2F5%5E%22%22%29%2BC\"\r\n" );
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document.write( "\"-3072cos%5E3%28theta%29%28+1%2F3-expr%281%2F5%29cos%5E2%28theta%29%29%2BC\"\r\n" );
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document.write( "\"-48%28sqrt%2816-x%5E2%29%29%5E3%281%2F3-expr%281%2F80%29%2816-x%5E2%29%29%2BC\"\r\n" );
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document.write( "\"-48%2816-x%5E2%29%5E%28%223%2F2%22%29%281%2F3-expr%281%2F80%29%2816-x%5E2%29%29%2BC\"\r\n" );
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document.write( "Factor out 1/240\r\n" );
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document.write( "\"-48%2816-x%5E2%29%5E%28%223%2F2%22%29%281%2F240%29%2880-3%2816-x%5E2%29%29%2BC\"\r\n" );
document.write( "\"expr%28-1%2F5%29%2816-x%5E2%29%5E%28%223%2F2%22%29%2880-48%2B3x%5E2%29%2BC\"\r\n" );
document.write( "\"expr%28-1%2F5%29%2816-x%5E2%29%5E%28%223%2F2%22%29%2832%2B3x%5E2%29%2BC\"\r\n" );
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document.write( "That is the same answer as I got above using algebraic substitution.\r\n" );
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document.write( "Edwin
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