`int _0^(pi/4) (xsinx)/(cos^3(x))dx`
First the indefinite integral has to be computed. The above integral can be refined as:
`int xtanxsec^2x dx`
Now applying integration by parts:
`int uv'=uv-int u'v `
`u=x, u'=1, v'=sec^2xtanx, v=(sec^2x)/2`
`=x(sec^2x)/2- int 1(sec^2x)/2 dx`
`=(xsec^2x)/2- int (sec^2x)/2 dx`
`=(xsec^2x)/2-(tanx)/2+C`
Computing the boundaries:
`int_0^(pi/4) (xsinx)/(cos^3x) dx=(pi-2)/4-0=(pi-2)/4`
Therefore,...
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`int _0^(pi/4) (xsinx)/(cos^3(x))dx`
First the indefinite integral has to be computed. The above integral can be refined as:
`int xtanxsec^2x dx`
Now applying integration by parts:
`int uv'=uv-int u'v `
`u=x, u'=1, v'=sec^2xtanx, v=(sec^2x)/2`
`=x(sec^2x)/2- int 1(sec^2x)/2 dx`
`=(xsec^2x)/2- int (sec^2x)/2 dx`
`=(xsec^2x)/2-(tanx)/2+C`
Computing the boundaries:
`int_0^(pi/4) (xsinx)/(cos^3x) dx=(pi-2)/4-0=(pi-2)/4`
Therefore, the answer is `(pi-2)/4` .
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