Question

Iff(x) =fotsint dt, thenf,(r) is(A) cosx x sin x(C) x cosx10.(B) x sinx(D) sinx +x cosx

Answer 1

The function is given as,

f( x ) =  ∫ 0 x t sintdt  

From the formula of integration by parts,

∫ g( r )  h( r )dr=g( r ) ∫ h( r )dr − ∫ ( g'( r ) ∫ h( r ) dr ) dr

Take g( r )=t and h( r )=sint ,

f( x ) =  ∫ 0 x t sin tdt            = [ t ∫ sin tdt − ∫ ( d dt t × ∫ sin t dt )dt ] 0 x = [ t×( −cost )− ∫ ( costdt ) ] 0 x = [ −t cost + sint ] 0 x

Simplify further,

f( x )= −x cosx+ sinx   =sinx −xcosx 

We have to differentiate to get the result,

f'( x )=  d dx ( sinx −x cosx ) =cosx−( x  d( cosx ) dx +cosx dx dx ) =cosx+x sinx−cosx =xsinx

Hence, option (B) is correct.