Question

quien me puede ayudar con esa integral

este es el resultado:

3^4x(3.3^4x+8.3^2x+6)/24ln3 + C

Answer 1

$\displaystyle \int\left(3^{2x}+3^{4x}\right)^2~dx=\int \left(e^{2x\ln 3}+e^{4x\ln 3}\right)^2~dx\\ \\ \\ \int\left(3^{2x}+3^{4x}\right)^2~dx=\int e^{4x\ln 3}+2e^{6x\ln 3}+e^{8x\ln 3}dx\\ \\ \\ \int\left(3^{2x}+3^{4x}\right)^2~dx=\dfrac{e^{4x\ln 3}}{4\ln 3}+2\cdot\dfrac{e^{6x\ln 3}}{6\ln 3}+\dfrac{e^{8x\ln 3}}{8\ln 3}+C$


$\displaystyle \int\left(3^{2x}+3^{4x}\right)^2~dx=\dfrac{3^{4x}}{4\ln 3}+\dfrac{3^{6x}}{3\ln 3}+\dfrac{3^{8x}}{8\ln 3}+C\\ \\ \\ \int\left(3^{2x}+3^{4x}\right)^2~dx=3^{4x}\left(\dfrac{1}{4\ln 3}+\dfrac{3^{2x}}{3\ln 3}+\dfrac{3^{4x}}{8\ln 3}\right)+C\\ \\ \\ \\ \boxed{\int\left(3^{2x}+3^{4x}\right)^2~dx=\dfrac{3^{4x}\left(6+8\cdot3^{2x}+3\cdot 3^{4x}\right)}{24\ln 3}+C}$