bài 5) `\int`$2^{x}$ ($3^{x}$+$4^{2x}$) dx=.... `\int`$2^{3x}$ .$3^{2x}$ dx=..... `\int`$\frac{5^{x}+1}{5^{x}}$ dx=........ `\int`$\frac{9^{x}-4}{3^{x}-2}$ dx=..... `\int`$\frac{8^{x}-1}{2^{x}-1}$ dx=............

Đáp án+Giải thích các bước giải:

$\displaystyle \int 2^x(3^x+4^{2x}) \, dx\\ =\displaystyle \int 2^x(3^x+2^{4x}) \, dx\\ =\displaystyle \int (6^x+2^{5x}) \, dx\\ =\displaystyle \int (6^x+32^x) \, dx\\ =\dfrac{6^x}{\ln 6}+\dfrac{32^x}{\ln 32} +C\\ \displaystyle \int 2^{3x}.3^{2x} \, dx\\ =\displaystyle \int 8^x.9^x \, dx\\ =\displaystyle \int 72^x \, dx\\ =\dfrac{72^x}{\ln 72}+C\\ \displaystyle \int \dfrac{5^x+1}{5^x}\, dx\\ =\displaystyle \int \left(1+\dfrac{1}{5^x}\right)\, dx\\ =\displaystyle \int \left(1+\left(\dfrac{1}{5}\right)\right)^x\, dx\\ =x+\dfrac{\left(\dfrac{1}{5}\right)^x}{\ln \dfrac{1}{5}}\\ =x-\dfrac{5^{-x}}{\ln 5}+C\\ \displaystyle \int \dfrac{9^x-4}{3^x-2}\, dx\\ \displaystyle \int \dfrac{(3^x)^2-2^2}{3^x-2}\, dx\\ \displaystyle \int \dfrac{(3^x-2)(3^x+2)}{3^x-2}\, dx\\ \displaystyle \int (3^x+2) \, dx\\ =\dfrac{3^x}{\ln 3}+2x+C\\ \displaystyle \int \dfrac{8^x-1}{2^x-1}\, dx\\ =\displaystyle \int \dfrac{(2^x)^3-1}{2^x-1}\, dx\\ =\displaystyle \int \dfrac{(2^x-1)((2^x)^2+2^x+1)}{2^x-1}\, dx\\ =\displaystyle \int ((2^x)^2+2^x+1)\, dx\\ =\displaystyle \int  (4^x+2^x+1)\, dx\\ =\dfrac{4^x}{\ln 4}+\dfrac{2^x}{\ln 2}+x+C.$