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प्रश्न
Integrate the following functions w.r.t. x:
sin (log x)
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उत्तर
Le I = `int sin (logx)x dx`
Put log x = t
∴ x = et
∴ dx = et 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 = – et cos t + et sin t – I
∴ 2I = et (sin t – cos t)
∴ `I = e^t/(2) (sin t - cos t) + c`
= `x/(2) [sin (logx) - cos (logx)] + c`.
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