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

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

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

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

- x + 2 tan (x) + 2 sec (x) + C

Solution steps

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

Expand

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

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

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

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

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

= tan (x) + 2 sec (x) + tan (x) - x

Simplify

= - x + 2 tan (x) + 2 sec (x)

Add a constant to the solution

= - x + 2 tan (x) + 2 sec (x) + C

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