Given:
The function is:
[tex]\int(2\tan x\cos x+7)dx[/tex]
Find-:
Evaluate the expression
Explanation-:
Use integration property is:
[tex]\int(a+b)dx=\int adx+\int bdx[/tex]
So the expression is:
[tex]\begin{gathered} =\int(2\tan x\cos x+7)dx \\ \\ =\int2\tan x\cos xdx+\int7dx \\ \\ =2\int\tan(x)\cos(x)dx+7\int1dx \end{gathered}[/tex]
Simplify the expression is:
[tex]\begin{gathered} =2\int\tan(x)\cos(x)+7\int1dx \\ \\ =2\int\frac{\sin x}{\cos x}\times\cos(x)dx+7\int1dx \\ \\ =2\int\sin(x)dx+7\int1dx \end{gathered}[/tex]
Use integration formula is:
[tex]\begin{gathered} \int\sin x=-\cos x \\ \\ \int1dx=x \end{gathered}[/tex]
So, the value is:
[tex]\begin{gathered} =2\int\sin(x)dx+7\int1dx \\ \\ =-2\cos x+7x+C \end{gathered}[/tex]
