Question

Integrate the following functions w.r.t. x:

sin (log x)

Answer 1

Le I  = `int sin (logx)x dx`

Put log x = t
∴ x = e^t
∴ dx = e^t dt

∴ I = `int sin t xx e^t dt`

= `int e^t sin t dt`

= `e^t int sin t dt - int [d/dt (e^t) int sin t dt] dt`

= `e^t (- cos t) - int e^t (- cos t) dt`

= `-e^t cos t + int e^t cos t dt`

= `- e^t cos t + e^t int cos t dt - int [d/dt (e^t) int cos t dt] dt`

= `- e^t cos t + e^t sin t - int e^t sin t dt`

∴ I = – e^t cos t + e^t sin t – I
∴ 2I = e^t (sin t – cos t)

∴ `I  = e^t/(2) (sin t - cos t) + c`

= `x/(2) [sin (logx) - cos (logx)] + c`.