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

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

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integral of (sin(2x)-2)/(cos^2(x))

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

- ln (cos^{2} (x)) - 2 tan (x) + C

Solution steps

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

Expand

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

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

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

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

= - ln (cos^{2} (x)) - 2 tan (x)

Add a constant to the solution

= - ln (cos^{2} (x)) - 2 tan (x) + C

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