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

1 Answer
May 28, 2018

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