What is the integral of sin(2t)(cot(2t)+1) ?

The integral of sin(2t)(cot(2t)+1) is 1/2 sin(2t)-1/2 cos(2t)+C

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integral of sin(2t)(cot(2t)+1)

\int sin (2 t) (cot (2 t) + 1) d t

\frac{1}{2} sin (2 t) - \frac{1}{2} cos (2 t) + C

Solution steps

\int sin (2 t) (cot (2 t) + 1) d t

Rewrite using trig identities

= \int - 1 + 2 cos^{2} (t) + sin (2 t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d t + \int 2 cos^{2} (t) d t + \int sin (2 t) d t

\int 1 d t = t

\int 2 cos^{2} (t) d t = t + \frac{1}{2} sin (2 t)

\int sin (2 t) d t = - \frac{1}{2} cos (2 t)

= - t + t + \frac{1}{2} sin (2 t) - \frac{1}{2} cos (2 t)

Simplify

= \frac{1}{2} sin (2 t) - \frac{1}{2} cos (2 t)

Add a constant to the solution

= \frac{1}{2} sin (2 t) - \frac{1}{2} cos (2 t) + C

\frac{d}{d x} (∣ x - 1 ∣^{3})

x \to 0 lim (∣ x - 1 ∣)

x \to 5 lim (2 x 5)

\frac{\partial}{\partial x} (e^{2 x} \cdot 2)

f (x) = (2 x - 7)^{4} (x^{2} + x + 1)^{5}

What is the integral of sin(2t)(cot(2t)+1) ?
The integral of sin(2t)(cot(2t)+1) is 1/2 sin(2t)-1/2 cos(2t)+C