What is the integral of x*cos(5x) ?

The integral of x*cos(5x) is 1/25 (5xsin(5x)+cos(5x))+C

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integral of x*cos(5x)

\int x \cdot cos (5 x) d x

\frac{1}{25} (5 x sin (5 x) + cos (5 x)) + C

Solution steps

\int x cos (5 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{25} d u

= \int \frac{u cos ( u )}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= \frac{1}{25} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{25} (u sin (u) - (- cos (u)))

Substitute back

u = 5 x

= \frac{1}{25} (5 x sin (5 x) - (- cos (5 x)))

Simplify

= \frac{1}{25} (5 x sin (5 x) + cos (5 x))

Add a constant to the solution

= \frac{1}{25} (5 x sin (5 x) + cos (5 x)) + C

x \to 0 lim (\frac{1}{1 + x ^{2}})

\frac{d}{d x} ((x^{- 2} + x)^{3} (4 x)^{- 2 x})

x y^{^{'}} = 4 x^{4} cos (2 x) + 3 y

x \to 0 lim (x sin (\frac{π}{x}))

y = 5 x - x^{2}, y = 2 x, (3, 6)

What is the integral of x*cos(5x) ?
The integral of x*cos(5x) is 1/25 (5xsin(5x)+cos(5x))+C