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

The integral of (x-1)cos(2x) is 1/4 (2xsin(2x)+cos(2x))-1/2 sin(2x)+C

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

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

\frac{1}{4} (2 x sin (2 x) + cos (2 x)) - \frac{1}{2} sin (2 x) + C

Solution steps

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

Expand

(x - 1) cos (2 x) : x cos (2 x) - cos (2 x)

= \int x cos (2 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 x cos (2 x) d x - \int cos (2 x) d x

\int x cos (2 x) d x = \frac{1}{4} (2 x sin (2 x) + cos (2 x))

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

= \frac{1}{4} (2 x sin (2 x) + cos (2 x)) - \frac{1}{2} sin (2 x)

Add a constant to the solution

= \frac{1}{4} (2 x sin (2 x) + cos (2 x)) - \frac{1}{2} sin (2 x) + C

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

What is the integral of (x-1)cos(2x) ?
The integral of (x-1)cos(2x) is 1/4 (2xsin(2x)+cos(2x))-1/2 sin(2x)+C