If y=f(x)=∫0x(x−t)6.sint dt then which of following are true.

y=∫0x(x−(x−t))6.sin(x−t)dty=∫0xt6.(sinx.cost−cosx.sint)y=sinx.∫0xt6cost dt−cosx.∫0xt6y''+y=x6,y'(0)=0

If y=f(x)=∫0x(x−t)6.sint dt then which of following are true.

y=∫0x(x−(x−t))6.sin(x−t)dty=∫0xt6.(sinx.cost−cosx.sint)y=sinx.∫0xt6cost dt−cosx.∫0xt6y''+y=x6,y'(0)=0

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If $y = f (x) = \int_{0}^{x} {(x - t)}^{6} . \sin t d t$ then which of following are true.

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$y^{''} + y = x^{6}$

$y^{'} (0) = 0$

$y'' - y = x^{6}$

$y^{'} (0) = 10$

answer is A, B.

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$y = \int_{0}^{x} {(x - (x - t))}^{6} . \sin (x - t) d t$

$\begin{array}{l} y = \int_{0}^{x} t^{6} . (\sin x . \cos t - \cos x . \sin t) \\ y = \sin x . \int_{0}^{x} t^{6} \cos t d t - \cos x . \int_{0}^{x} t^{6} \\ y^{''} + y = x^{6}, y^{'} (0) = 0 \end{array}$

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If y=f(x)=∫0x(x−t)6.sint dt then which of following are true.

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If $y = f (x) = \int_{0}^{x} {(x - t)}^{6} . \sin t d t$ then which of following are true.