What is the integral of (xcos(3x)) ?

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

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integral of (xcos(3x))

\int (x 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

x \to - 2 lim (\frac{2 x ^{3} + 9 x ^{2} + 12 x + 4}{- x ^{3} - 2 x ^{2} + 4 x + 8})

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

\int_{- 1}^{1} (x^{3} - x) d x

\int_{0}^{c} 9 e^{3 x} d x

\frac{\partial}{\partial x} (3 sin (2 x y))

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