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# integrate x*sinxdx

## 2 Answers

8 years ago

∫xsinx dx = x∫sinx - ∫d/dx(x)(∫sinxdx) dx .........(taking x as first function and sinx as second);

= x(-cosx) - ∫d/dx(x) (-cosx) dx

= -xcosx + ∫cosx dx

= -xcosx + sinx + C

hence ∫xsinx dx = -xcosx + sinx + C

8 years ago

or you can write

∫xsinx dx = Im ∫x e^(ix) dx

= Im (x∫e^(ix) dx - ∫d/dx(x) ∫e^(ix)dx dx)

= Im (xe^(ix)/i - ∫e^(ix)/i dx)

= Im (-ixe^(ix) + e^(ix)/i)

= Im (-ixe^(ix) - ie^(ix))

= Im (-ie^(ix) (x-1))

= Im (-i(cosx + isinx)(x-1))

= Im ((x-1)sinx -i(x-1)cosx)

= (x-1)sinx

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