What is the integral of-xcos(x+pi/3) ?

The integral of-xcos(x+pi/3) is -xsin(x+pi/3)-cos(x+pi/3)+C

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integral of-xcos(x+pi/3)

\int - x cos (x + \frac{π}{3}) d x

- x sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) + C

Solution steps

\int - x cos (x + \frac{π}{3}) d x

Take the constant out:

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

= - \int x cos (x + \frac{π}{3}) d x

Apply Integration By Parts

: x sin (x + \frac{π}{3}) - \int sin (x + \frac{π}{3}) d x

= - (x sin (x + \frac{π}{3}) - \int sin (x + \frac{π}{3}) d x)

\int sin (x + \frac{π}{3}) d x = - cos (x + \frac{π}{3})

= - (x sin (x + \frac{π}{3}) - (- cos (x + \frac{π}{3})))

Simplify

= - x sin (x + \frac{π}{3}) - cos (x + \frac{π}{3})

Add a constant to the solution

= - x sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) + C

\int x^{2} e^{- 2 x^{3}} d x

\int 2 [1 - cos^{2} (2 x)] d x

\int_{- 1}^{1} π (2 - 2 x^{2})^{2} d x

y^{^{''}} + 2 y^{^{'}} - 15 y = 4 e^{- t}

What is the integral of-xcos(x+pi/3) ?
The integral of-xcos(x+pi/3) is -xsin(x+pi/3)-cos(x+pi/3)+C