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education/math/MATH1210 (calc 1)/Integrals.md

@@ -89,5 +89,10 @@ $$\int_a^b f(x) dx = F(b) - F(a)$$
 2. Let $f$ be a continuous function on $[a, b]$ and let $x$ be a point in $[a, b]$.
 $$ F(x) = \int_a^x f(t)dt \Rightarrow F'(x) = f(x) $$
 $$ \dfrac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) * g'(x)* $$
+## Examples
+> Finding the derivative of an integral
 $$ \dfrac{d}{dx} \int_2^{7x} \cos(t^2) dt = cos((7x)^2) * 7 = 7\cos(49x^2)$$
-$$ \dfrac{d}{dx}\int_0^{\ln{x}}\tan(t) = \tan(\ln(x))*\dfrac{1}{x} $$
+> Finding the derivative of an integral
+$$ \dfrac{d}{dx}\int_0^{\ln{x}}\tan(t) = \tan(\ln(x))*\dfrac{1}{x} $$
+> $x$ and $t$ notation
+$$ F(x) = \int_4^x 2t \space dt = t^2 \Big|_4^x  = x^2 - 16$$