First Fundamental x a d f t dt f x dx 1 Derivative of an integral

First fundamental x a d f t dt f x dx 1 derivative of

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First Fundamental Theorem:
( 29 ( 29 x a d f t dt f x dx = 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. First Fundamental Theorem:
cos x d t dt dx π - cos x = 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. 29 29 29 The long way: First Fundamental Theorem:
2 0 1 1+t x d dt dx 2 1 1 x = + 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.
2 0 cos x d t dt dx ( 29 2 2 cos d x x dx ( 29 2 cos 2 x x ( 29 2 2 cos x x The upper limit of integration does not match the derivative, but we could use the chain rule .
5 3 sin x d t t dt dx The lower limit of integration is not a constant, but the upper limit is. 5 3 sin x d t t dt dx - 3 sin x x - We can change the sign of the integral and reverse the limits .
2 2 1 2 x t x d dt dx e + Neither limit of integration is a constant. 2 0 0 2 1 1 2 2 x t t x d dt dt dx e e + + + It does not matter what constant we use! 2 2 0 0 1 1 2 2 x x t t d dt dt dx e e - + + 2 2 1 1 2 2 2 2 x x x e e - + + (Limits are reversed.) (Chain rule is used.) 2 2 2 2 2 2 x x x e e = - + + We split the integral into two parts .
The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of , and if F is any antiderivative of f on , then [ ] , a b ( 29 ( 29 ( 29 b a f x dx F b F a = - [ ] , a b (Also called the Integral Evaluation Theorem ) We already know this! To evaluate an integral, take the anti-derivatives and subtract. π

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