Evaluate ∫π0e|cosx|(2sin(12cosx)+3cos(12cosx))sinxdx
Let I=∫π0e|cosx|(2sin(12cosx)+3cos(12cosx))sinxdx
⇒I=∫π0e|cosx|.sinx.2sin(12cosx)dx+∫π0e|cosx|.3cos(12cosx).sinxdx
⇒I=I1+I2 ...(i)
Using ∫2a0f(x)dx=⎧⎨⎩0,f(2a−x)=−f(x)2∫a0f(x)dx,f(2a−x)=+f(x)
Here, I1=0 [∵f(π−x)=−f(x)] ...(ii)
I2=6∫π20ecosx.sinx.cos(12cosx)dx [∵f(π−x)=f(x)]
Put cosx=t⇒−sinxdx=dt
When x=0⇒t=1
When x=π2⇒t=0
I2=6∫10et.cos(t2)dt
Applying integration by parts
=6[etcos(t2)+12∫etsin(t2)dt]10
Again applying integration by parts, we get
=6[etcos(t2)+12(etsint2−∫et2cost2dt)]10
I2=6[etcost2+12etsint2]10−I24
I2+I24=6[etcost2+12etsint2]10
⇒I2=245(ecos(12)+e2sin(12)−1) ...(iii)
From eq.(i), we get
I=245(ecos(12)+e2sin(12)−1)