Differentiating integral functions

**I-Think** · February 8th 2011, 09:07 PM

Need to make sure I'm getting this right

Is

$A) \frac{d}{dx}\int_3^{e{^x^2}} ln(t^3) dt=2x^3e^x^2<br />$
and

$B) \frac{d}{dx}\int_a^x \frac{f(t}{t^2}=\frac{-2f(x)}{x}$

**FernandoRevilla** · February 8th 2011, 09:20 PM

Originally Posted by I-Think

$A) \frac{d}{dx}\int_3^e^x^2 ln(t^3) dt=2x^3e^x^2<br />$

It is not clear what the integrand function and limits are.

$B) \frac{d}{dx}\int_a^x \frac{f(t}{t^2}=\frac{-2f(x)}{x}$

$\dfrac{d}{dx}\displaystyle\int_a^x \dfrac{f(t)}{t^2}dt=\dfrac{f(x)}{x^2}$

Fernando Revilla

**I-Think** · February 8th 2011, 10:08 PM

For A), integrating $ln(t^3)$ between $e^x^2$ and $3$

**FernandoRevilla** · February 8th 2011, 11:11 PM

Using the Fundamental Theorem of Calculus an the Chain's Rule:

$\dfrac{d}{dx}\displaystyle\int_a^{g(x)} f(t)dt=f[g(x)]g'(x)$

Let us see what do you obtain.

Fernando Revilla

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