Nguyên hàm của hàm số 1 + tan^2x/2

Giải thích các bước giải:

$\int 1+\dfrac{\tan^2x}2dx$ 

$=\int \dfrac{1}{2}+\dfrac{\tan^2x+1}2dx$ 

$=\int \dfrac{1}{2}+\dfrac{\dfrac{\sin^2x}{\cos^2x}+1}2dx$ 

$=\int \dfrac{1}{2}+\dfrac{\dfrac{\sin^2x+\cos^2x}{\cos^2x}}2dx$ 

$=\int \dfrac{1}{2}+\dfrac{\dfrac{1}{\cos^2x}}2dx$ 

$=\int \dfrac{1}{2}+\dfrac{1}{2\cos^2x}dx$ 

$=\dfrac{1}{2}x+\dfrac{1}{2}\tan x+C$ 

$f(x)=1+\dfrac{1}{2}\tan^2x$

$=1+\dfrac{1}{2}\Big(1+\tan^2x\Big)-\dfrac{1}{2}$

$=\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{\cos^2x}$

$\int f(x)dx$

$=\int \dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{\cos^2x}dx$

$=\dfrac{1}{2}x+\dfrac{1}{2}\tan x+C$