answer:To solve this integral, we apply the method of substitution. Let u = 3x^2 - 2x + 2, which implies du = 6x - 2 dx. Since we have 3x - 1 in the numerator, we can write 3x - 1 as frac{1}{2}(6x - 2), and thus frac{1}{2}du = (3x - 1)dx. With this substitution, the integral becomes: int frac{3x - 1}{u^6} left(frac{1}{2} duright) Simplifying further, we get: frac{1}{2} int frac{1}{u^6} du Now, integrate with respect to u: frac{1}{2} left(-frac{1}{5} frac{1}{u^5}right) + C = -frac{1}{10u^5} + C Finally, we express the answer back in terms of x: int frac{3x - 1}{(3x^2 - 2x + 2)^6} dx = -frac{1}{10(3x^2 - 2x + 2)^5} + C

answer:Given: Initial velocity of the electron is {eq}vec v =(10.5 hat j + 12.5 hat k) km/s = (10500 hat j +12500 hat k) m/s {/eq}. Acceleration acting is {eq}vec a =1.80times 10^{12 } m/s^2hat i {/eq}. Magnetic field present is {eq}B=400 mu T = 400 times 10^{-6} T hat i {/eq} Mass and charge of the electron is {eq}m=9.11 times 10^{-31}kg and q= -1.6 times 10^{-19}C {/eq} Lorentz force on the electron; {eq}F = F_E+F_M {/eq} According to Newton's second law of motion {eq}F= ma {/eq}, therefore; {eq}begin{align*} ma &=q vec E + q (vec B times vec v) ma&=q(vec E + vec v times vec B) 9.11 times 10^{-31} times 1.80times 10^{12 }hat i &=- 1.6 times 10^{-19} (vec E + (10500 hat j + 12500 hat k) times (400 times 10^{-6} hat i ) ) 16.4 times 10^{-19} hat i &=- 1.6 times 10^{-19} (vec E -4.2 hat k+5 hat j) -10.25 hat i &= vec E -4.2 hat k+5 hat j vec E &=- 10.25 hat i -5 hat j + 4.2 hat k end{align*} {/eq} Thus we have {eq}E_x =-10.25 N/C E_y =-5 N/C E_z = 4.2 N/C {/eq}

answer:Using the formula to calculate bond's yield to maturity: {eq}YTM = dfrac{C + dfrac{F-P}{n}}{dfrac{F+P}{2}} {/eq} Values from the question are: Annual Coupon (C) = 85 Face Value (F) = 1000 Price (P) = 1120 Time to maturity (n) = 15 years Substituting the values into the formula: {eq}YTM = dfrac{85 +dfrac{1000 - 1120}{15}}{dfrac{1000 + 1120}{2}} {/eq} {eq}YTM = dfrac{85 +dfrac{-120}{15}}{dfrac{2120}{2}} {/eq} {eq}YTM = dfrac{85 + -8}{1060} {/eq} {eq}YTM = dfrac{77}{1060} {/eq} {eq}YTM = 0.0726 {/eq} Therefore, the yield to maturity of the bond is 7.26%.

answer:Gross Domestic Product (GDP) is a key economic indicator that represents the total market value of all final goods and services produced within a country's borders in a specific time period, typically a year. It excludes intermediate goods and services as well as those produced by foreign entities. GDP can be calculated using three main methods: 1. **Income Approach**: Summing up the incomes of all factors of production, such as wages, rent, profits, and capital gains. 2. **Expenditure Approach**: Adding up the total spending on final goods and services by consumers, businesses, the government, and net exports (exports minus imports). 3. **Net Value-Added Approach**: Calculating the difference between the value of output and the value of intermediate consumption for each sector of the economy. These approaches provide a comprehensive view of a country's economic activity and are used to gauge its overall economic health.