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

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

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integral of x(cos(2x))

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

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

Solution steps

\int x cos (2 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{4} d u

= \int \frac{u cos ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{4} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= \frac{1}{4} (u sin (u) - (- cos (u)))

Substitute back

u = 2 x

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

Simplify

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

Add a constant to the solution

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

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