query: Proof that integrating f(x) between a and b is the same as integrating f(a+b-x) over the same bounds?
positive_passages.docid: 16109
positive_passages.text: Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
negative_passages.docid: 132747
negative_passages.text: How can one show that $ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$?
negative_passages.docid: 70512
negative_passages.text: Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$
negative_passages.docid: 7903
negative_passages.text: Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$
negative_passages.docid: 40047
negative_passages.text: Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?
negative_passages.docid: 106799
negative_passages.text: Proof Verification: Converse of Intermediate Value Theorem
negative_passages.docid: 98271
negative_passages.text: Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$
negative_passages.docid: 80199
negative_passages.text: Do we have always $f(A \cap B) = f(A) \cap f(B)$?
negative_passages.docid: 157608
negative_passages.text: One-sided limits of $f'(x)$ at a point vs. one-sided limits of the difference quotient at that point
negative_passages.docid: 59063
negative_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 134551
negative_passages.text: Comparison between integrals
negative_passages.docid: 22303
negative_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 30204
negative_passages.text: Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
negative_passages.docid: 53236
negative_passages.text: Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$
negative_passages.docid: 28798
negative_passages.text: Proving injection and surjection with functions $F:C^B\to C^A$, $F(f)=f\circ\varphi$, $\varphi: A\to B$
negative_passages.docid: 136966
negative_passages.text: Riemann integral proof $\int^b_a f(x) \, dx>0$
negative_passages.docid: 43733
negative_passages.text: $|f'(x)| \le g(x)$ implies $|f(b) - f(a)| \le \int_a^b g(x) dx$, without assuming $f'$ to be integrable.