$If space straight f left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x straight t space sint space dt comma space write space the space value space of space straight f apostrophe left parenthesis straight x right parenthesis$

Since differentiation operation is the inverse operation of integration, we have
$straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x space sinx Let space straight f left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x straight t space sint space dt$
Let us do this by integration by parts.
Therefore assume u = t; du = dt

$integral sint space dt space space equals space integral dv minus cost space equals straight v Therefore comma space straight f left parenthesis straight x right parenthesis space equals space open square brackets straight t left parenthesis negative cost right parenthesis close square brackets subscript 0 superscript straight x space minus space integral subscript 0 superscript straight x left parenthesis negative cos space straight t right parenthesis space dt straight f left parenthesis straight x right parenthesis space equals space minus straight x space cosx space plus space sinx space plus straight C$
Differentiating the above function with respect to x,

$straight f apostrophe left parenthesis straight x right parenthesis space equals space minus open square brackets straight x left parenthesis negative sinx right parenthesis minus sinx close square brackets space plus space sinx space equals space straight x space sinx$

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