What is the integral of 18xcos(3x) ?

The integral of 18xcos(3x) is 2(3xsin(3x)+cos(3x))+C

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integral of 18xcos(3x)

\int 18 x cos (3 x) d x

2 (3 x sin (3 x) + cos (3 x)) + C

Solution steps

\int 18 x cos (3 x) d x

Take the constant out:

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

= 18 \cdot \int x cos (3 x) d x

Apply u-substitution

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

= 18 \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

= 18 \cdot \frac{1}{9} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= 18 \cdot \frac{1}{9} (u sin (u) - \int sin (u) d u)

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

= 18 \cdot \frac{1}{9} (u sin (u) - (- cos (u)))

Substitute back

u = 3 x

= 18 \cdot \frac{1}{9} (3 x sin (3 x) - (- cos (3 x)))

Simplify

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

= 2 (3 x sin (3 x) + cos (3 x))

Add a constant to the solution

= 2 (3 x sin (3 x) + cos (3 x)) + C

x y + y^{2} - x^{2} \cdot \frac{d y}{d x} = 0

\int x + x^{2} d x

\int 8 cot (e^{4 x}) e^{4 x} d x

x \to 3 lim (x - 3)

What is the integral of 18xcos(3x) ?
The integral of 18xcos(3x) is 2(3xsin(3x)+cos(3x))+C