What is the integral of ((1+sin(x))/(cos^2(x))) ?

The integral of ((1+sin(x))/(cos^2(x))) is tan(x)+sec(x)+C

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integral of ((1+sin(x))/(cos^2(x)))

\int (\frac{1 + sin ( x )}{cos ^{2} ( x )}) d x

tan (x) + sec (x) + C

Solution steps

\int \frac{1 + sin ( x )}{cos ^{2} ( x )} d x

Expand

\frac{1 + sin ( x )}{cos ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} + \frac{sin ( x )}{cos ^{2} ( x )}

= \int \frac{1}{cos ^{2} ( x )} + \frac{sin ( x )}{cos ^{2} ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{cos ^{2} ( x )} d x + \int \frac{sin ( x )}{cos ^{2} ( x )} d x

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

\int \frac{sin ( x )}{cos ^{2} ( x )} d x = sec (x)

= tan (x) + sec (x)

Add a constant to the solution

= tan (x) + sec (x) + C

What is the integral of ((1+sin(x))/(cos^2(x))) ?
The integral of ((1+sin(x))/(cos^2(x))) is tan(x)+sec(x)+C