5.3 The Fundamental Theorem of Calculus: Difference between revisions
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:1. FTC #1 <br> | :1. FTC #1 <br> | ||
: <math>\frac{d}{dx}\left[\int_{a(x)}^{b(x)}f(t)dt\right]=b'(x)\ | : <math>\frac{d}{dx}\left[\int_{a(x)}^{b(x)}f(t)dt\right]=b'(x)\cdot f(b(x))-a'(x)\cdot f(a(x))</math><br><br> | ||
:2. FTC #2 <br> | :2. FTC #2 <br> | ||
: <math>\int_{a}^{b}f(x)dx=F(b)-F(a)</math><br> Where <math>\frac{d}{dx}\left[F(x)\right]=f(x)</math>. <math>F(x)</math> is called the <b>antiderivative</b> of <math>f(x)</math> <br> | : <math>\int_{a}^{b}f(x)dx=F(b)-F(a)</math><br> Where <math>\frac{d}{dx}\left[F(x)\right]=f(x)</math>. <math>F(x)</math> is called the <b>antiderivative</b> of <math>f(x)</math> <br> | ||
![{\displaystyle {\frac {d}{dx}}\left[\int _{a(x)}^{b(x)}f(t)dt\right]=b'(x)\cdot f(b(x))-a'(x)\cdot f(a(x))}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d5c751ba93084f4b7b1f5b5a354da424d15e9507)
![{\displaystyle {\frac {d}{dx}}\left[F(x)\right]=f(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3a5307d913d3c9aa3aa8e81954173adc0f657140)
