vault backup: 2025-04-01 10:08:23

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

@@ -94,5 +94,7 @@ $$ \dfrac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) * g'(x)* $$
 $$ \dfrac{d}{dx} \int_2^{7x} \cos(t^2) dt = cos((7x)^2) * 7 = 7\cos(49x^2)$$
 > 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
+> $x$ and $t$ notation *(note: the bar notation is referred to as "evaluated at")*
 $$ F(x) = \int_4^x 2t \space dt = t^2 \Big|_4^x  = x^2 - 16$$
+> $x$ in top and bottom
+$$ \dfrac{d}{dx} \int_{2x}^{3x} \sin(t) dt = \dfrac{d}{dx} -\cos(t)\Big|_{2x}^{3x} = \dfrac{d}{dx} (-\cos(3x) + cos(2x) = 3\sin(3x) - 2\sin(2x) $$