What is the integral of cos(2x)-(1-4/pi x) ?

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

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integral of cos(2x)-(1-4/pi x)

\int cos (2 x) - (1 - \frac{4}{π} x) d x

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

Solution steps

\int cos (2 x) - (1 - \frac{4}{π} 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 1 - \frac{4}{π} x d x

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

\int 1 - \frac{4}{π} x d x = x - \frac{2 x ^{2}}{π}

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

Simplify

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

Add a constant to the solution

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

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