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

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

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integral of cos(3x)x

\int cos (3 x) x d x

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

Solution steps

\int cos (3 x) 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

y = x^{x^{x}}

\int_{0}^{t} 1 d t

\frac{1}{4 π ( 3 + 2 ) ^{2}} (- 11)

x \to 1 - lim (1)

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