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Improper Integrals:
Definition. Let f be a function which is continuous on the closed interval [a, infinity). We define
If this limit exists and is finite then we say that the integral  is convergent; otherwise, we say that the integral is divergent. |
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Example
Definition. Let f be a function which is continuous on the closed interval (infinity, b]. We define
If this limit exists and is finite then we say that the integral  is convergent; otherwise, we say that the integral is divergent. |
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Integration by Parts and by Partial Fractions
A very useful technique for evaluating integrals is Integration by Parts:
It is derived from the product formula for derivatives. Sometimes it is more convenient to express this formula using differentials:
Example:
Arc Length:
Definition. Let x = h(t) and y = g(t) be parametric functions such that the derivatives h' and g' are continuous on the closed interval [r, s]. The arc length of the graph of this system of parametric functions from t = r to t = s is the integral
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