What is the integral of (tan^2(x)+cot^2(x)+2) ?

The integral of (tan^2(x)+cot^2(x)+2) is -2cot(2x)+C

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integral of (tan^2(x)+cot^2(x)+2)

\int (tan^{2} (x) + cot^{2} (x) + 2) d x

- 2 cot (2 x) + C

Solution steps

\int tan^{2} (x) + cot^{2} (x) + 2 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 tan^{2} (x) d x + \int cot^{2} (x) d x + \int 2 d x

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

\int cot^{2} (x) d x = - x - cot (x)

\int 2 d x = 2 x

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

Simplify

- x + tan (x) - x - cot (x) + 2 x : - 2 cot (2 x)

= - 2 cot (2 x)

Add a constant to the solution

= - 2 cot (2 x) + C

\int \frac{40}{x ^{2} + 25} d x

\frac{d}{d x} (3 x^{2} cos (x))

\frac{d y}{d x} = (y - 1)^{2} + 0.01

h (x) = (- 3 x + 2)^{4} (- 2 x + 3)^{3}

\int t^{2} sin (β) t d t

What is the integral of (tan^2(x)+cot^2(x)+2) ?
The integral of (tan^2(x)+cot^2(x)+2) is -2cot(2x)+C