\[
\int 4e^{-7x}dx \\
U = -7x \\
dU = -7dx \\
dx = \frac{dU}{-7} \\
4\int e^U\frac{dU}{-7} \\
\frac{4}{-7}\int e^UdU \\
\frac{4}{-7} e^U + C \\
\frac{4}{-7} e^{-7x} + C
\]
\[ \frac{dN}{dt} =\frac{3150}{t^4}-220 \\ dN = (\frac{3150}{t^4}-220)dt \\ N = \int \frac{3150}{t^4}dt-\int 220dt \\ N = N_0 - \frac{3150}{3t^3} - 220t \\ N = N_0 - \frac{3150}{3t^3} - 220t \\ N(1) = N_0 - \frac{1050}{1^3} - 220(1) \\ N_0 = 6530 + 1050 + 220 \\ N_0 = 7800 \\ N = 7800 - \frac{1050}{t^3} - 220t \]
\[ A = \int_{4.5}^{8.5} 2x-9 dx \\ A = [x^2 - 9x]|_{4.5}^{8.5} \\ A = [8.5^2-9*8.5]-[4.5^2-9*4.5] \\ A = 16 \]
\[ y = x^2 -2x-2 \\ y =x+2 \\ A = \int_{-1}^{4}x+2 dx -\int_{-1}^{4}x^2 -2x-2 dx\\ A = \frac{1}{2}x^2|_{-1}^{4} +2x|_{-1}^{4} -[\frac{1}{3}x^3 - x^2 -2x]|_{-1}^{4} \\ A = -[\frac{1}{3}x^3 - \frac{3}{2}x^2 -4x]|_{-1}^{4} \\ A = [\frac{3}{2}x^2 +4x -\frac{1}{3}x^3]|_{-1}^{4} \]
## [1] 20.83333
\[ C = 8.25r + \frac{3.75x}{2} \\ C = 8.25r+\frac{206.25}{r} \\ C' = 8.25 - \frac{206.25}{r^2} \\ C' = 0 \\ r = \sqrt{\frac{206.25}{8.25}}\\ \]
5 Orders per Year
\[ \int ln(9x)*x^6dx \\ U = ln(9x) \\ dU = \frac{1}{x}dx \\ dV = x^6dx \\ V = \frac{1}{7}x^7 \\ \int UdV = UV - \int VdU \\ \frac{1}{7}ln(9x)x^7 - \frac{1}{7}\int x^6dx \\ \frac{1}{7}x^7[ln(9x) - \frac{1}{7}] \]
\[ F(x) = \int_{1}^{e^6} f(x)dx = 1 \\ f(x) = \frac{1}{6x} \\ F(x) = \int_{1}^{e^6} \frac{1}{6x}dx \\ F(x) = \frac{1}{6} \int_{1}^{e^6} \frac{1}{x}dx \\ F(x) = \frac{1}{6} ln(x)|_1^{e^6} \\ F(x) = \frac{1}{6} [ln(e^6) - ln(1)] \\ F(x) = \frac{1}{6} [6-0] = 1 \]