1.
\(\int\left(\dfrac{1}{5x+3}+cos3x\right)dx=\dfrac{1}{5}ln\left|5x+3\right|+\dfrac{1}{3}sin3x+C\)
\(\int\left(e^{2x}+2\right)^3dx=\int\left(e^{6x}+6e^{4x}+12e^{2x}+8\right)dx=\dfrac{1}{6}e^{6x}+\dfrac{3}{2}e^{4x}+6e^{2x}+8x+C\)
\(\int x\left(x^2+7\right)^2dx=\dfrac{1}{2}\int\left(x^2+7\right)^2d\left(x^2+7\right)=\dfrac{1}{6}\left(x^2+7\right)^3+C\)
\(\int e^x\left(2020+\dfrac{2021e^{-x}}{x^6}\right)dx=\int\left(2020.e^x+\dfrac{2021}{x^6}\right)dx=2020e^x-\dfrac{2021}{5x^5}+C\)
\(\int\left(x+1\right)\left(x+2\right)\left(x+3\right)dx=\int\left(x^3+6x^2+11x+6\right)dx=\dfrac{x^4}{4}+2x^3+\dfrac{11}{2}x^2+6x+C\)
\(\int\dfrac{x^3+6x+2}{x}dx=\int\left(x^2+6+\dfrac{2}{x}\right)dx=\dfrac{x^3}{3}+6x+2ln\left|x\right|+C\)
\(\int\sqrt[3]{2x+1}dx=\dfrac{1}{2}\int\left(2x+1\right)^{\dfrac{1}{3}}d\left(2x+1\right)=\dfrac{3}{8}\left(2x+1\right)^{\dfrac{4}{3}}+C=\dfrac{3}{8}\sqrt[3]{\left(2x+1\right)^4}+C\)
2.
\(F\left(x\right)=\int\left(2+cos^2x-cos2x\right)dx=\int\left(2+\dfrac{1}{2}+\dfrac{1}{2}cos2x-cos2x\right)dx\)
\(=\int\left(\dfrac{5}{2}-\dfrac{1}{2}cos2x\right)dx=\dfrac{5}{2}x-\dfrac{1}{4}sin2x+C\)
Do\(F\left(0\right)=1\Rightarrow\dfrac{5}{2}.0-\dfrac{1}{4}sin0+C=1\Rightarrow C=1\)
\(\Rightarrow F\left(x\right)=\dfrac{5}{2}x-\dfrac{1}{4}sin2x+1\)
\(\Rightarrow F\left(\dfrac{\pi}{2}\right)=\dfrac{5}{2}.\dfrac{\pi}{2}-\dfrac{1}{4}sin\left(\pi\right)+1=\dfrac{5\pi}{4}+1\)
3.
\(B\left(t\right)=\int B'\left(t\right)dt=\int\dfrac{3000}{\left(1+0,2t\right)^2}dt=\int\dfrac{15000}{\left(1+0,2t\right)^2}d\left(1+0,2t\right)\)
\(=-\dfrac{15000}{1+0,2t}+C\)
\(B\left(0\right)=10000\Rightarrow-\dfrac{15000}{1}+C=10000\Rightarrow C=25000\)
\(\Rightarrow B\left(t\right)=25000-\dfrac{15000}{1+0,2t}\)
\(\Rightarrow B\left(7\right)=25000-\dfrac{15000}{1+0,2.7}=...\)
