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Integrals

Question

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$Find colon space integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared space end root space dx.$

Solution

straight I space equals integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx Let space left parenthesis straight x space plus 3 right parenthesis space equals space straight A straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis space plus space straight B space space space... space left parenthesis straight i right parenthesis rightwards double arrow space space straight x space plus space 3 space equals space straight A left parenthesis negative 2 straight x minus 4 right parenthesis space plus space straight B rightwards double arrow space straight x plus 3 space equals space minus 2 Ax minus 4 straight A plus straight B therefore space minus 2 straight A space space equals space 1 rightwards double arrow space straight A space equals space minus 1 half minus 4 straight A space plus space straight B space equals space 3 rightwards double arrow space minus 4 space open parentheses negative 1 half close parentheses plus space straight B space equals 3 rightwards double arrow space space straight B equals 1 therefore comma straight I space equals integral open square brackets negative 1 half straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis plus 1 close square brackets space square root of 3 minus 4 straight x minus straight x squared end root space dx equals space space 1 half integral straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx space plus space integral square root of 3 minus 4 straight x minus straight x squared minus 4 plus 4 dx end root equals space minus 1 half fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 divided by 2 end fraction space plus space integral square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root dx equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared dx end root equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root space plus space 7 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator square root of 7 end fraction close parentheses space space plus space straight C

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