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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{tan ^{7} ( x )}{sec ^{8} ( x )} d x

\int \frac{a}{u ^{2} + a ^{2}} d u

\int \frac{2}{9 + x ^{2}} d x

\int (3 x^{3} + 6 x)^{3} (3 x^{2} + 2) d x

\int t t + 2 d t

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