हल कीजिए-
`sin ^(-1)((dy)/(dx))=x+y.`

Explanation

यहाँ `sin ^(-1) ((dy)/(dx))=x+y`
`implies (dy)/(dx) =sin (x+y)" "...(1)`
माना `x+y=v" "...(2)`
x के सापेक्ष अवकलन करने पर,
`1+(dy)/(dx)=(dv)/(dx) `
`implies (dy)/(dx) =(dv)/(dx)-1" "...(3)`
समी (1),(2) और (3)
`(dv)/(dx) =1+sin v`
`implies (dv)/(dx) =1+sin v`
`implies(dv)/(1+sin v)=dx`
समाकलन करने पर,
`int(dv)/(1+sinv) =int dx`
`=int ((1)/(1+sin v)xx (1-sin v)/(1-sin v))dv =x+C`
`implies int (1- sin v)/(1-sin^(2)v)dv=x+C`
`implies int (1- sin v)/(cos^(2)v)dv=x+C`
`impliesint((1)/(cos ^(2)v)-(sin v)/(cos ^(2)v))dv=x+c`
`impliesintsec^(2) vdv-int tan v sec vdv=x+C.`
`impliestan v- sec v=x+C`
`implies tan (x+y) -sec (x+y) =x+C.`