What is the integral of x*cos(3x) ?

The integral of x*cos(3x) is 1/9 (3xsin(3x)+cos(3x))+C

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integral of x*cos(3x)

\int x \cdot cos (3 x) d x

\frac{1}{9} (3 x sin (3 x) + cos (3 x)) + C

Solution steps

\int x cos (3 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{9} d u

= \int \frac{u cos ( u )}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= \frac{1}{9} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{9} (u sin (u) - (- cos (u)))

Substitute back

u = 3 x

= \frac{1}{9} (3 x sin (3 x) - (- cos (3 x)))

Simplify

= \frac{1}{9} (3 x sin (3 x) + cos (3 x))

Add a constant to the solution

= \frac{1}{9} (3 x sin (3 x) + cos (3 x)) + C

(- 2, - 5), (- 4, 4)

x \to 4 lim (x^{4} e^{- x^{3}})

x \to 1 + lim ((ln (x))^{1 - x})

\int (x - 1) 1 + e^{2 y} d y

\int (\frac{1}{2 x + 5}) d x

What is the integral of x*cos(3x) ?
The integral of x*cos(3x) is 1/9 (3xsin(3x)+cos(3x))+C