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Evaluate the following integral: int0^(p...

Evaluate the following integral: `int_0^(pi//4)sin2x sin\ 3x\ dx`

Text Solution

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Given:`int_0^(pi/4)sin2x sin\ 3x\ dx`
`I=int_0^(pi/4)sin2x sin\ 3x\ dx`
`I=1/2int_0^(pi/4)sin2x sin\ 3x\ dx`
As `2sinA.sinB=cos(A-B)-cos(A+B)`
`I=1/2 int_0^(pi/4) [cos(3x-2x)-cos(3x+2x)]dx`
`I=1/2 int_0^(pi/4) cosxdx-1/2 int_0^(pi/4) cos5xdx`
`I=1/2[sinx]_0^(pi/4)-1/2[sin5x]_0^(pi/4)`
`I=1/2[sin(pi/4)]-1/10[sin5(pi/4)]`
...
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