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Question Number 122434 by benjo_mathlover last updated on 17/Nov/20

If f(x) = ∫_a ^( (∫_a ^x^3 (dt/(1+sin^2 t)) )) ((dt/(1+sin^2 t))) then f ′(x) ?

$$\:{If}\:{f}\left({x}\right)\:=\:\int_{{a}} ^{\:\left(\underset{{a}} {\overset{{x}^{\mathrm{3}} } {\int}}\:\frac{{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} {t}}\:\right)} \left(\frac{{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} {t}}\right) \\ $$$${then}\:{f}\:'\left({x}\right)\:? \\ $$

Answered by liberty last updated on 17/Nov/20

f ′(x) = (((x^3 )′)/((1+sin^2 (x^3 )).(1+sin^2 (∫_1 ^x^3 (dt/(1+sin^2 t)))) )) f ′(x) = ((3x^2 )/((1+2sin^2 (x^3 )).(1+sin^2 (∫_1 ^x^3 (dt/(1+sin^2 t))))))

$$\:\mathrm{f}\:'\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{x}^{\mathrm{3}} \right)'}{\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\right).\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\underset{\mathrm{1}} {\overset{\mathrm{x}^{\mathrm{3}} } {\int}}\:\frac{\mathrm{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{t}}\right)\right)\:} \\ $$$$\:\mathrm{f}\:'\left(\mathrm{x}\right)\:=\:\frac{\mathrm{3x}^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{2sin}\:^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\right).\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\underset{\mathrm{1}} {\overset{\mathrm{x}^{\mathrm{3}} } {\int}}\:\frac{\mathrm{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{t}}\right)\right)} \\ $$

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