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If u and v are two functions of x then p...

If u and v are two functions of x then prove that: `intuv dx =uint vdx - int [(du)/(dx)int vdx] dx`

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`"Let" int vdx=w," Then "(dw)/(dx)=v`
By the rule for the derivtive of the product of two functions.
`(d)/(dx)(uw)=u(dw)/(dx)+w(du)/(dx)=uv+w(du)/(dx)`
`therefore` by the definition of indefinite integral.
`int (uv+w(du)/(dx))dx=uw`
`therefore int uv+w(du)/(dx))dx=uw`
`therefore int uv dx+int (w(du)/(dx))dx=uw`
`therefore int uv" "dx=uw-int (w(du)/(dx))dx`
`therefore int uv" "dx=udx-int ((du)/(dx).int vdx)dx.`
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