5.3 The Fundamental Theorem of Calculus/17

FTC #1

$G(x)=f^{\prime }(x)$ or in other words ${\frac {d}{dx}}$ of $\int \limits _{a(x)}^{b(x)}F(x)dx$ is $\ b^{\prime }(x)*f(b(x))-a^{\prime }(x)*f(a(x))$

$y=\int \limits _{1-3x}^{1}{\frac {x^{3}}{(1+u^{2})}}dx$

so $y=\int \limits _{1-3x}^{1}{\frac {1}{(1+u^{2})}}x^{3},dx$

using the formula we get y=$(0)*f(1)-(-3)*f(1-3x)$

which is equal to $(3)*f(1-3x)$

which is=$3*(1-3x)^{3}*{\frac {1}{(1+(1-3x)^{2})}}$

or simplified to ${\frac {3*(1-3x)^{3}}{(1+(1-3x)^{2})}}$

FTC #2 $\int \limits _{a}^{b}f(x)dx$ is equal to $F(b)-F(a)$ Where F is the antiderivitive of f such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F^\prime=f}