We would like to evaluate
(a) Compute f ( x ) = ∫ − 1 x sin ( t 2 ) 2 t d t . {\displaystyle f(x)=\int _{-1}^{x}\sin(t^{2})2t\,dt.}
(b) Find f ′ ( x ) . {\displaystyle f'(x).}
(c) State the Fundamental Theorem of Calculus.
(d) Use the Fundamental Theorem of Calculus to compute d d x ( ∫ − 1 x sin ( t 2 ) 2 t d t ) {\displaystyle {\frac {d}{dx}}{\bigg (}\int _{-1}^{x}\sin(t^{2})2t\,dt{\bigg )}} without first computing the integral.
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without first computing the integral.