answer:Given {eq}displaystyle int (2x+1) sqrt{x^2+x+1} dx{/eq} We use the method of substitution to solve the integral. We use the substitution {eq}x^2+x+1=u{/eq} to solve the integral. Differentiate both sides with respect to {eq}x{/eq} {eq}begin{align} 2x+1 &=dfrac{du}{dx} (2x+1) dx & =du end{align}{/eq} Applying this substitution to the integral, we have: {eq}begin{align} int (2x+1) sqrt{x^2+x+1} dx &=int sqrt{u} du &=dfrac{u^{frac{1}{2}+1}}{frac{1}{2}+1}+C &left [ because int x^ndx=dfrac{x^{n+1}}{n+1} right ] &=dfrac{2}{3}u^{frac{3}{2}}+C &=dfrac{2}{3}(x^2+x+1)^{frac{3}{2}}+C end{align}{/eq} Therefore, {eq}color{blue}{boxed{int (2x+1) sqrt{x^2+x+1} dx=dfrac{2}{3}(x^2+x+1)^{frac{3}{2}}+C}}{/eq}

answer:120,000

answer:Butterfly blood is colorless and clear. Unlike humans and other vertebrates, butterflies and other insects do not have red blood cells or hemoglobin. Instead, they use a respiratory pigment called hemocyanin to transport oxygen throughout their bodies. Hemocyanin is a copper-based protein that turns blue when oxygenated and colorless when deoxygenated. Since butterfly blood does not contain hemoglobin, it does not have the characteristic red color associated with mammalian blood.

answer:Given the radius of the circle, r, is (15.9 ± 0.5) m. (a) The area, A, can be calculated as: [ A = pi r^2 ] The area is: [ A = pi (15.9)^2 approx 794.23 , m^2 ] To find the uncertainty in the area, we use the chain rule: [ frac{Delta A}{A} = 2 frac{Delta r}{r} ] [ Delta A = A cdot 2 frac{0.5}{15.9} approx 49.95 , m^2 ] So, the area of the circle is (794.23 ± 49.95) m^2. (b) The circumference, c, is: [ c = 2pi r ] The circumference is: [ c = 2pi (15.9) approx 99.90 , m ] For the uncertainty in the circumference, we apply the same method: [ frac{Delta c}{c} = frac{Delta r}{r} ] [ Delta c = c cdot frac{0.5}{15.9} approx 3.14 , m ] Thus, the circumference of the circle is (99.90 ± 3.14) m.