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

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

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

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

x - sin^{2} (x) + C

Solution steps

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

Expand

(sin (x) - cos (x))^{2} : sin^{2} (x) - 2 sin (x) cos (x) + cos^{2} (x)

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

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

\int 2 sin (x) cos (x) d x = sin^{2} (x)

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

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) - sin^{2} (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

Simplify

\frac{1}{2} (x - \frac{1}{2} sin (2 x)) - sin^{2} (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : x - sin^{2} (x)

= x - sin^{2} (x)

Add a constant to the solution

= x - sin^{2} (x) + C

x^{3} - 27 x

x^{2} y^{^{'}} - y x = 0

\frac{d}{d x} (6 x^{4} - 2 x + 7)

\frac{1}{( s - 1 ) ^{2}}

\int_{0}^{1} k x d x

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