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

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

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

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

cos (x) + tan (x) + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int - sin (x) + sec^{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 sin (x) d x + \int sec^{2} (x) d x

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

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

= - (- cos (x)) + tan (x)

Simplify

= cos (x) + tan (x)

Add a constant to the solution

= cos (x) + tan (x) + C

\int \frac{e ^{2 x}}{81 + e ^{4 x}} d x

e^{2 x}, e^{x}, x = ln (3)

\frac{d}{d x} (x sin (\frac{9}{x}))

\int 3 sin (t) + 2 cos (t) + 2 d t

x + 2, \frac{1}{x + 1}, [0, 2]

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