These are some rules of definite integrals that are pretty self explanatory, and have most of which been discussed before.
1. \(\int_{a}^{a} f(x) \ dx = 0 \)
2. \(\int_{a}^{b} f(x) \ dx = - \int_{b}^{a} f(x) \ dx \)
3. \(\int_{a}^{b} k \cdot f(x) \ dx = k \int_{a}^{b} f(x) \ dx \ \ \ \text{For any constant, } k \)
4. \(\int_{a}^{b} f(x) \pm g(x) \ dx = \int_{a}^{b} f(x) \ dx \pm \int_{a}^{b} g(x) \ dx \)
5. \(\int_{a}^{b} f(x) \ dx + \int_{b}^{c} f(x) \ dx = \int_{a}^{c} f(x) \ dx \)
1) \(\int_{6}^{12} f(x) \ dx = 7 \text{,} \ \ \ \int_{6}^{10} f(x) \ dx = 4 \)
a. \(\displaystyle \int_{12}^{6} -3 f(x) \ dx \)
\(\begin{aligned} \ \ \ \ \ \ &= -3 \int_{12}^{6} f(x) \ dx \\[5pt] &= 3 \int_{6}^{12} f(x) \ dx \\[5pt] &= 3 \times 7 \\[5pt] &= 21 \end{aligned} \)
b. \(\int_{6}^{8} f(x) \ dx + \int_{8}^{10} f(x) \ dx \)
\(\begin{aligned} \ \ \ \ \ \ &= \int_{6}^{10} f(x) dx \\[5pt] &= 4 \end{aligned} \)
\(\displaystyle \cfrac{d}{dx} \left( \int_{a}^{x} f(v) \ dv \right) = f(x) \)
\(2) \displaystyle \ \frac{d}{dt} \int_{-2}^{t} (3x-2)^3 \ dx \)
\(\ \ \ \ \ \ = (3t-2)^3 \)
\(\int_{a}^{b} f'(x) \ dx = f(b) - f(a) \)
3) \(\ \displaystyle \int_{-1}^{3} 3x^2 + 2x^2 \ dx \)
\(\begin{aligned} \int 3x^2 + 2x^2 \ dx &= x^3 + \frac{2x^2}{3} + c \\[5pt] &= ((3)^3 + \frac{2(3)^2}{3} + c) - ((-1)^3 + \frac{2(-1)^2}{3} + c) \\[5pt] \int_{-1}^{3} 3x^2 + 2x^2 \ dx &= (27 + 6) - (-1 + \frac{2}{3}) \\[5pt] &= \frac{100}{3} \end{aligned} \)
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