Question

Evaluate: int(1+2sinx/cos^2x)dx?

Answer 1

`I=tan(x)+2sec(x)+C`

Explanation:

I assume you want to integrate

`I=int(1+2sin(x))/(cos^2(x))dx`

Splitting into two terms

`I=int1/(cos^2(x))dx+int(2sin(x))/(cos^2(x))dx`

First integral
Remember `color(blue)((tan(x))'=sec^2(x)`

`I_1=int1/(cos^2(x))dx`

`color(white)(I)=intsec^2(x)dx`

`color(white)(I)=tan(x)+C_1`

Second integral
Make a substitution `color(red)(u=cos(x)=>du=-sin(x)dx`

`I_1=int(2sin(x))/(cos^2(x))dx`

`color(white)(I_1)=-2int1/u^2du larr color(red)("The substitution"`

`color(white)(I_1)=2/u+C_2`

`color(white)(I_1)=2/cos(x)+C_2`

`color(white)(I_1)=2sec(x)+C_2`

Combining these

`I=tan(x)+2sec(x)+C`