Question

integral de e^3x + ln (3x) + pi

Answer 1

$\displaystyle \int\,e^{3x}+\ln(3x)+\pi\,dx\\\\i)~~~~\int\,e^{3x}+\ln(3x)+\pi\,dx=\int e^{3x}dx+\int \ln(3x)dx+\int \pi dx\\\\\\~~~I)~~~\int e^{3x}dx\implies 3x=u\implies \frac{du}{3}=dx\\\\~~~II)~~\int e^{3x}dx=\frac{1}{3}\int e^udu=\frac{1}{3}e^{3x}+c_1\\\\~~III)~~\int \ln(3x)dx=x\ln(3x)-\int \frac{x}{x}dx=x\ln3x-x+c_2\\\\~~IV)~\int \pi dx=\pi x+c_3\\\\ii)~~~\int e^{3x}+\ln(3x)+\pi dx=\frac{1}{3}e^{3x}+x\ln|3x|-x+\pi x+(c_1+c_2+c_3)\\\\iii)~~\int e^{3x}+\ln(3x)+\pi x dx=\boxed{\boxed{\frac{1}{3}e^{3x}+x\ln|3x|-x+\pi x+C}}$

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