What is the integral of-2/3 xcos(3x) ?

The integral of-2/3 xcos(3x) is -2/27 (3xsin(3x)+cos(3x))+C

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integral of-2/3 xcos(3x)

\int - \frac{2}{3} x cos (3 x) d x

- \frac{2}{27} (3 x sin (3 x) + cos (3 x)) + C

Solution steps

\int - \frac{2}{3} x cos (3 x) d x

Take the constant out:

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

= - \frac{2}{3} \cdot \int x cos (3 x) d x

Apply u-substitution

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

= - \frac{2}{3} \cdot \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{2}{3} \cdot \frac{1}{9} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= - \frac{2}{3} \cdot \frac{1}{9} (u sin (u) - \int sin (u) d u)

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

= - \frac{2}{3} \cdot \frac{1}{9} (u sin (u) - (- cos (u)))

Substitute back

u = 3 x

= - \frac{2}{3} \cdot \frac{1}{9} (3 x sin (3 x) - (- cos (3 x)))

Simplify

- \frac{2}{3} \cdot \frac{1}{9} (3 x sin (3 x) - (- cos (3 x))) : - \frac{2}{27} (3 x sin (3 x) + cos (3 x))

= - \frac{2}{27} (3 x sin (3 x) + cos (3 x))

Add a constant to the solution

= - \frac{2}{27} (3 x sin (3 x) + cos (3 x)) + C

f (t) = t e^{t} cot (t)

f (x) = \frac{5}{x}, x = - 1

arctan (\frac{1 - x}{1 + x})

x \to - \infty lim (81 x^{2} + x - 9 x)

n = 0 \sum \infty 9 (\frac{x}{7})^{n}

What is the integral of-2/3 xcos(3x) ?
The integral of-2/3 xcos(3x) is -2/27 (3xsin(3x)+cos(3x))+C