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The Second Fundamental Theorem of Calculus d x f t dt f x dx a The Fundamental Theorem of Calculus, Part 1 If f is continuous on a, b , then the function F x f t dt x a has a derivative at every point in a, b , and dF d x f t dt f x dx dx a Second Fundamental Theorem: d x f t dt f x dx a 1. Derivative of an integral. Second Fundamental Theorem: d x f t dt f x dx a 1. Derivative of an integral. 2. Derivative matches upper limit of integration. Second Fundamental Theorem: d x f t dt f x dx a 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. Second Fundamental Theorem: d x f t dt f x dx a New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: Second Fundamental Theorem: d x cos x cos t dt dx d x sin t dx d dx 0 sin x sin d sin x dx cos x 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. d x 1 1 dt 2 dx 0 1+t 1 x2 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. d x cos t dt dx 0 2 d 2 cos x x dx 2 cos x 2 2 x 2 x cos x 2 The upper limit of integration does not match the derivative, but we could use the chain rule. d 5 3t sin t dt dx x d x 3t sin t dt dx 5 3x sin x The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits. d x 1 dt t dx 2 x 2 e 2 Neither limit of integration is a constant. We split the integral into two parts. 0 d x2 1 1 dt dt 0 t t 2x 2 e dx 2 e It does not matter what constant we use! 2x d x2 1 1 dt dt 0 t t 0 2e dx 2 e (Limits are reversed.) 1 1 2x 2 2x 2 (Chain xrule 2 is used.) 2x 2x x2 2e 2e 2e 2e The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of a, b , and if F is any antiderivative of f on a, b , then f x dx F b F a b a (Also called the Integral Evaluation Theorem) To evaluate an integral, take the anti-derivatives and subtract.