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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

\int sin^{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

n = 0 \sum \infty 2^{- 2 - 2 n}

π \cdot c \cdot \frac{cos ( \frac{π x}{L} )}{L}

\int 6 x^{2} sin (\frac{1}{2} x) d x

\frac{x ^{2} - 1}{x + 3}

e^{2 - t}

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