How do I find the integral intcos(x)ln(sin(x))dx ?
1 Answer
Jul 30, 2014
=ln(sin(x))(sin(x)-1)+c , wherec is a constantExplanation
=intcos(x)ln(sin(x))dx let's
sin(x)=t ,=> cos(x)dx=dt
=intln(t)*1dt Using Integration by Parts,
int(I)(II)dx=(I)int(II)dx-int((I)'int(II)dx)dx where
(I) and(II) are functions ofx , and(I) represents which will be differentiated and(II) will be integrated subsequently in the above formulaSimilarly following for the problem,
=ln(t)int1*dt-int((lnt)'intdt)dt
=tln(t)-int1/t*tdt
=tln(t)-t+c , wherec is a constant
=t(ln(t)-1)+c , wherec is a constant
=sin(x)(lnsin(x)-1)+c , wherec is a constant
