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int sin^(-1/2)x cos^ (-7/2)x dx=...

`int sin^(-1/2)x cos^ (-7/2)x dx=`

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`int ((1* (cos x)^(1/2))/(sin x)^(1/2) * 1/(cos x)^(7/2)*1/(cos x)^(1/2)) dx`
`= int(tanx)^(-1/2) (sec^2 x)^2 dx`
`= int((tan x)^(-1/2) (1 + tan^2 x)sec^2 x dx `
let `t= tanx `
`dt = sec^2 x dx = dt`
`= int t^(-1/2) (1 + t^2) dt`
`= int t^(-1/2) dt + int t^(3/2) dt`
`= (t^(-1/2 + 1))/(-1/2 + 1) + (t^(3/2 + 1))/(3/2 + 1) + c`
...
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