Evaluate   integral left parenthesis space sin x minus x cos x right parenthesis space d x space divided by space x space left parenthesis x plus sin x right parenthesis

Asked by ahuja8087 | 29th Jan, 2017, 10:56: PM

Expert Answer:

begin mathsize 16px style L e t space I equals integral fraction numerator sinx minus xcosx over denominator straight x left parenthesis straight x plus sinx right parenthesis end fraction dx
space straight I equals integral fraction numerator sinx plus straight x minus straight x minus xcosx over denominator straight x left parenthesis straight x plus sinx right parenthesis end fraction dx
space straight I equals integral fraction numerator left parenthesis sinx plus straight x right parenthesis minus left parenthesis straight x plus xcosx right parenthesis over denominator straight x left parenthesis straight x plus sinx right parenthesis end fraction dx
space straight I equals integral open square brackets 1 over straight x minus fraction numerator left parenthesis straight x plus xcosx right parenthesis over denominator straight x left parenthesis straight x plus sinx right parenthesis end fraction close square brackets dx
space straight I equals integral 1 over straight x dx minus integral fraction numerator left parenthesis straight x plus xcosx right parenthesis over denominator straight x left parenthesis straight x plus sinx right parenthesis end fraction dx
space straight I equals ln open vertical bar straight x close vertical bar minus integral fraction numerator 1 plus cosx over denominator straight x plus sinx end fraction dx
Let space straight x plus sinx space equals space straight t rightwards double arrow left parenthesis 1 plus cosx right parenthesis space dx space equals space dt
So comma space straight I equals ln open vertical bar straight x close vertical bar minus integral dt over straight t
rightwards double arrow straight I equals ln open vertical bar straight x close vertical bar minus ln open vertical bar straight t close vertical bar plus straight C
rightwards double arrow straight I equals ln open vertical bar straight x close vertical bar minus ln open vertical bar straight x plus sinx close vertical bar plus straight C

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Answered by Rebecca Fernandes | 30th Jan, 2017, 09:56: AM