Find begin mathsize 20px style integral fraction numerator 3 e to the power of x minus 5 e to the power of negative x end exponent over denominator 9 e to the power of x minus 16 e to the power of negative x end exponent end fraction end style

Asked by sunil2791 | 4th Jul, 2017, 06:31: AM

Expert Answer:

begin mathsize 16px style straight I equals integral fraction numerator 3 straight e to the power of straight x minus 5 straight e to the power of negative straight x end exponent over denominator 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent end fraction dx
Numerator equals straight A cross times Denominator plus straight B cross times straight d over dx open parentheses Denominator close parentheses
3 straight e to the power of straight x minus 5 straight e to the power of negative straight x end exponent equals straight A open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses plus straight B straight d over dx open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses
3 straight e to the power of straight x minus 5 straight e to the power of negative straight x end exponent equals straight A open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses plus straight B open parentheses 9 straight e to the power of straight x plus 16 straight e to the power of negative straight x end exponent close parentheses........... left parenthesis straight I right parenthesis
3 straight e to the power of straight x minus 5 straight e to the power of negative straight x end exponent equals open parentheses 9 straight A plus 9 straight B close parentheses straight e to the power of straight x plus open parentheses negative 16 straight A plus 16 straight B close parentheses straight e to the power of negative straight x end exponent
Comparing space both space sides comma
9 straight A plus 9 straight B equals 3 rightwards double arrow 3 straight A plus 3 straight B equals 1
minus 16 straight A plus 16 straight B equals negative 5
Solving space these space two space equations space simultaneously space you space get space
straight A equals 31 over 96 comma space straight B equals 1 over 96
Put space these space in space equation space left parenthesis straight I right parenthesis
integral open parentheses fraction numerator 31 open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses over denominator 96 open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses end fraction plus fraction numerator 9 straight e to the power of straight x plus 16 straight e to the power of negative straight x end exponent over denominator 96 open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses end fraction close parentheses dx
integral open parentheses 31 over 96 plus 1 over 96 fraction numerator 9 straight e to the power of straight x plus 16 straight e to the power of negative straight x end exponent over denominator open parentheses 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close parentheses end fraction close parentheses dx
31 over 96 straight x plus 1 over 96 log open vertical bar 9 straight e to the power of straight x minus 16 straight e to the power of negative straight x end exponent close vertical bar plus straight c end style

Answered by Sneha shidid | 4th Jul, 2017, 10:22: AM