5.3 The Fundamental Theorem of Calculus/17

FTC #1

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

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

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

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 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 \frac{3*(1-3x)^3}{(1+(1-3x)^2)}}

FTC #2 (not done yet)

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 \int\limits_{a}^{b}f(x)dx} is equal to 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(b)-F(a)} Where F is the antiderivative 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}

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

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

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