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Asked by jain.pradeep | 10th Jan, 2020, 10:21: AM

Expert Answer:

To find the integration of e-3x cos3x
integral straight e to the power of negative 3 straight x end exponent space cos cubed straight x space dx
equals integral straight e to the power of negative 3 straight x end exponent space open square brackets fraction numerator cos 3 straight x plus 3 cosx over denominator 4 end fraction close square brackets space dx
equals 1 fourth open square brackets integral straight e to the power of negative 3 straight x end exponent space cos 3 straight x space dx plus integral straight e to the power of negative 3 straight x end exponent cosxdx close square brackets
equals 1 fourth open square brackets straight I subscript 1 plus straight I subscript 2 close square brackets
straight I subscript 1 equals integral straight e to the power of negative 3 straight x end exponent space cos 3 straight x space dx equals straight e to the power of negative 3 straight x end exponent integral cos 3 straight x space dx space minus space integral open square brackets straight d over dx open parentheses straight e to the power of negative 3 straight x end exponent close parentheses integral cos 3 straight x space dx close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space fraction numerator sin 3 straight x over denominator 3 end fraction space minus space integral open square brackets straight e to the power of negative 3 straight x end exponent open parentheses negative 3 close parentheses fraction numerator sin 3 straight x over denominator 3 end fraction close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space fraction numerator sin 3 straight x over denominator 3 end fraction space plus space integral open square brackets straight e to the power of negative 3 straight x end exponent space sin 3 straight x close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space fraction numerator sin 3 straight x over denominator 3 end fraction space plus space straight e to the power of negative 3 straight x end exponent open parentheses fraction numerator negative cos 3 straight x over denominator 3 end fraction close parentheses minus integral straight e to the power of negative 3 straight x end exponent open parentheses negative 3 close parentheses open parentheses fraction numerator negative cos 3 straight x over denominator 3 end fraction close parentheses dx
equals straight e to the power of negative 3 straight x end exponent space fraction numerator sin 3 straight x over denominator 3 end fraction space minus space straight e to the power of negative 3 straight x end exponent open parentheses fraction numerator cos 3 straight x over denominator 3 end fraction close parentheses minus integral straight e to the power of negative 3 straight x end exponent open parentheses cos 3 straight x close parentheses dx
equals straight e to the power of negative 3 straight x end exponent space fraction numerator sin 3 straight x over denominator 3 end fraction space minus space straight e to the power of negative 3 straight x end exponent open parentheses fraction numerator cos 3 straight x over denominator 3 end fraction close parentheses minus straight I subscript 1
rightwards double arrow straight I subscript 1 equals straight e to the power of negative 3 straight x end exponent over 6 open square brackets space sin 3 straight x space minus cos 3 straight x close square brackets plus straight C subscript 1
straight I subscript 2 equals integral straight e to the power of negative 3 straight x end exponent cosxdx equals straight e to the power of negative 3 straight x end exponent integral cosx space dx space minus space integral open square brackets straight d over dx open parentheses straight e to the power of negative 3 straight x end exponent close parentheses integral cosx space dx close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space sinx space minus space integral open square brackets straight e to the power of negative 3 straight x end exponent open parentheses negative 3 close parentheses sinx close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space sinx space plus 3 integral open square brackets straight e to the power of negative 3 straight x end exponent sinx close square brackets dx equals straight e to the power of negative 3 straight x end exponent space sinx space minus 3 straight e to the power of negative 3 straight x end exponent cosx minus 9 integral open square brackets straight e to the power of negative 3 straight x end exponent cosx close square brackets dx
equals straight e to the power of negative 3 straight x end exponent space sinx space minus 3 straight e to the power of negative 3 straight x end exponent cosx minus 9 straight I subscript 2
rightwards double arrow straight I subscript 2 equals fraction numerator straight e to the power of negative 3 straight x end exponent space over denominator 10 end fraction open square brackets sinx space minus 3 cosx close square brackets plus straight C subscript 2
rightwards double arrow integral straight e to the power of negative 3 straight x end exponent space cos cubed straight x space dx equals 1 fourth open square brackets straight e to the power of negative 3 straight x end exponent over 6 open square brackets space sin 3 straight x space minus cos 3 straight x close square brackets plus straight C subscript 1 plus fraction numerator 3 straight e to the power of negative 3 straight x end exponent space over denominator 10 end fraction open square brackets sinx space minus 3 cosx close square brackets plus straight C subscript 2 close square brackets
equals straight e to the power of negative 3 straight x end exponent over 120 open square brackets 5 space sin 3 straight x space minus 5 space cos 3 straight x space plus space 9 sinx space minus space 27 space cosx close square brackets plus straight C

Answered by Renu Varma | 10th Jan, 2020, 11:50: AM