Let f(x) be a continuous increasing function in the interval
a £ x £ b where 0 < a < b and 0 £ f(a) < f(b). Prove
the formula
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ó
õ
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f(b)
f(a)
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f-1(t)dt + |
ó
õ
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b
a
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f(x)dx = bf(b) -af(a). |
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By considering the function f(x)=x2 find the value of
ò14 Öt dt in two ways, both by evaluating the
integral directly and also by using the formula above.
Use the formula to evaluate ò01sin-1t dt.