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

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

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integral of 2x*cos(2x)

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

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

Solution steps

\int 2 x cos (2 x) d x

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \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

= 2 \cdot \frac{1}{4} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= 2 \cdot \frac{1}{4} (u sin (u) - \int sin (u) d u)

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

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

Substitute back

u = 2 x

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

Simplify

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

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

Add a constant to the solution

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

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