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

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

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

\int 2 (sin (x) - cos (x) + 1) d x

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

Solution steps

\int 2 (sin (x) - cos (x) + 1) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int sin (x) - cos (x) + 1 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

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

tan (5 x)

\int \frac{cos ( 4 x )}{2} d x

\int sec (x) tan (x) (\frac{x}{π}) d x

\frac{d y}{d t} + y = cos (2 t), y (0) = 5

\frac{d y}{d x} = \frac{1}{x y}

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