If F(x) = _ x^2 ^ x^3 t ,dt, , ,(x > 0), then F'(x) = - Vidyadip

MCQ

If $F(x) = \int_{{x^2}}^{{x^3}} {\log t\,dt,\,\,(x > 0),} $ then $F'(x) = $

✓
$(9{x^2} - 4x)\log x$
B
$(4x - 9{x^2})\log x$
C
$(9{x^2} + 4x)\log x$
D
None of these

✓

Answer

Correct option: A.

$(9{x^2} - 4x)\log x$

a
(a) $F(x) = \int_{{x^2}}^{{x^3}} {\log t\,dt} $

Applying Leibnitzaes theorem,

$F\,'(x) = \log {x^3}.\frac{d}{{dx}}{x^3} - \log {x^2}.\frac{d}{{dx}}{x^2}$

$ = 3\log x \cdot 3{x^2} - 2\log x \cdot 2x$

$ = (9{x^2} - 4x)\log x$.

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