query: Is $\int_{a}^{b} f(x)\,dx$ equal to $- \int_{b}^{a} f(x)\,dx$?
positive_passages.docid: 61495
positive_passages.text: Why does an integral change signs when flipping the boundaries?
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: 98271
negative_passages.text: Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$
negative_passages.docid: 133990
negative_passages.text: How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$
negative_passages.docid: 150756
negative_passages.text: Equality of integrals
negative_passages.docid: 15409
negative_passages.text: Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$.
negative_passages.docid: 55175
negative_passages.text: Is the function $f(x) = x^2 - 2x$ one-to-one?
negative_passages.docid: 80199
negative_passages.text: Do we have always $f(A \cap B) = f(A) \cap f(B)$?
negative_passages.docid: 133851
negative_passages.text: How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$?
negative_passages.docid: 104145
negative_passages.text: Finding the value of $3(\alpha-\beta)^2$ if $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all $f$
negative_passages.docid: 80710
negative_passages.text: Does $\int_{-1}^1 \frac 1 x dx$ equal zero?
negative_passages.docid: 132448
negative_passages.text: Prove that: $\int_0^{n\pi}f(\cos^2(x))\,dx=n\int_0^{\pi}f(\cos^2(x))\,dx?$
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.
negative_passages.docid: 97773
negative_passages.text: If $f$ is continuous with $f(x) = f(2x),f(1) = 3$, then what is $ \int_{-1}^{1}f(f(x))\,dx$?
negative_passages.docid: 89956
negative_passages.text: Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?
negative_passages.docid: 137664
negative_passages.text: What exactly is $F$ in $W = \int_{a}^{b} F dx$?
negative_passages.docid: 33542
negative_passages.text: When is $A- B = B- A$?