CBSE Class 12-science Answered

integrate dx/({x-b}*sqrt[(x-a)*(x-b)])

Asked by amanj | 08 May, 2014, 09:53: PM

Expert Answer

$integral fraction numerator d x over denominator open parentheses x minus b close parentheses square root of open parentheses x minus a close parentheses open parentheses x minus b close parentheses end root end fraction equals integral fraction numerator d x over denominator open parentheses x minus b close parentheses to the power of begin display style bevelled 3 over 2 end style end exponent open parentheses x minus a close parentheses end fraction L e t space t equals fraction numerator 1 over denominator square root of open parentheses x minus b close parentheses end root end fraction rightwards double arrow d t equals minus 1 half open parentheses x minus b close parentheses to the power of minus bevelled 3 over 2 end exponent d x rightwards double arrow minus 2 d t equals fraction numerator d x over denominator open parentheses x minus b close parentheses to the power of begin display style bevelled 3 over 2 end style end exponent end fraction N o w comma space w e space h a v e equals integral fraction numerator minus 2 d x over denominator square root of open parentheses begin display style 1 over t squared end style plus b minus a close parentheses end root end fraction equals integral fraction numerator minus 2 t d x over denominator square root of 1 plus t squared open parentheses b minus a close parentheses end root end fraction 1 plus t squared open parentheses b minus a close parentheses equals m rightwards double arrow 2 t open parentheses b minus a close parentheses d t equals d m equals minus fraction numerator 1 over denominator open parentheses b minus a close parentheses end fraction integral fraction numerator d m over denominator m to the power of begin display style bevelled 1 half end style end exponent end fraction equals minus fraction numerator 1 over denominator open parentheses b minus a close parentheses end fraction integral fraction numerator d m over denominator m to the power of begin display style bevelled 1 half end style end exponent end fraction equals minus fraction numerator 1 over denominator open parentheses b minus a close parentheses end fraction m to the power of bevelled 1 half end exponent equals minus fraction numerator 1 over denominator open parentheses b minus a close parentheses end fraction square root of 1 plus t squared open parentheses b minus a close parentheses end root equals minus fraction numerator 1 over denominator open parentheses b minus a close parentheses end fraction square root of 1 plus fraction numerator open parentheses b minus a close parentheses over denominator x minus b end fraction end root plus C$