What is the integral of 4xcos(4x) ?

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

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integral of 4xcos(4x)

\int 4 x cos (4 x) d x

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

Solution steps

\int 4 x cos (4 x) d x

Take the constant out:

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

= 4 \cdot \int x cos (4 x) d x

Apply u-substitution

= 4 \cdot \int \frac{u cos ( u )}{16} d u

Take the constant out:

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

= 4 \cdot \frac{1}{16} \cdot \int u cos (u) d u

Apply Integration By Parts

= 4 \cdot \frac{1}{16} (u sin (u) - \int sin (u) d u)

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

= 4 \cdot \frac{1}{16} (u sin (u) - (- cos (u)))

Substitute back

u = 4 x

= 4 \cdot \frac{1}{16} (4 x sin (4 x) - (- cos (4 x)))

Simplify

4 \cdot \frac{1}{16} (4 x sin (4 x) - (- cos (4 x))) : x sin (4 x) + \frac{1}{4} cos (4 x)

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

Add a constant to the solution

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

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