I=∫0π/43sinx+5cosx136sinxdx136sinx=A(3sinx+5cosx)+B(3cosx−5sinx)136=3A−5B...(i)0=5A+3B...(ii) 3 B=−5 A⇒B=−35 A136=3 A−5(−35 A)136=3 A+325 A136=334 A⇒A=34136×3=12 B=3−5(12)=−20 I=∫0π/43sinx+5cosxA(3sinx+5cosx)+∫0π/43sinx+5cosxB(3cosx−5sinx)=A(x)0π/4+B[ln(3sinx+5cosx)]0π/4=12(4π)−20ln(23+25)−ln(0+5)=3π−20ln42+20ln5=3π−20×25ln2+20ln5=3π−50ln2+20ln5