What is the integral of (2sin(w)-1)/(cos^2(w)) ?

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

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integral of (2sin(w)-1)/(cos^2(w))

\int \frac{2 sin ( w ) - 1}{cos ^{2} ( w )} d w

2 cos (w) + tan (w) (- 1 + 2 sin (w)) + C

Solution steps

\int \frac{2 sin ( w ) - 1}{cos ^{2} ( w )} d w

Apply Integration By Parts

: - \int 2 sin (w) d w + tan (w) (- 1 + 2 sin (w))

= - \int 2 sin (w) d w + tan (w) (- 1 + 2 sin (w))

\int 2 sin (w) d w = - 2 cos (w)

= - (- 2 cos (w)) + tan (w) (- 1 + 2 sin (w))

Simplify

= 2 cos (w) + tan (w) (- 1 + 2 sin (w))

Add a constant to the solution

= 2 cos (w) + tan (w) (- 1 + 2 sin (w)) + C

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