What is the integral of x/2 cos(5x) ?

The integral of x/2 cos(5x) is 1/50 (5xsin(5x)+cos(5x))+C

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integral of x/2 cos(5x)

\int \frac{x}{2} cos (5 x) d x

\frac{1}{50} (5 x sin (5 x) + cos (5 x)) + C

Solution steps

\int \frac{x}{2} cos (5 x) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x cos (5 x) d x

Apply u-substitution

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

= \frac{1}{2} \cdot \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}{2} \cdot \frac{1}{25} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= \frac{1}{2} \cdot \frac{1}{25} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{2} \cdot \frac{1}{25} (u sin (u) - (- cos (u)))

Substitute back

u = 5 x

= \frac{1}{2} \cdot \frac{1}{25} (5 x sin (5 x) - (- cos (5 x)))

Simplify

\frac{1}{2} \cdot \frac{1}{25} (5 x sin (5 x) - (- cos (5 x))) : \frac{1}{50} (5 x sin (5 x) + cos (5 x))

= \frac{1}{50} (5 x sin (5 x) + cos (5 x))

Add a constant to the solution

= \frac{1}{50} (5 x sin (5 x) + cos (5 x)) + C

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What is the integral of x/2 cos(5x) ?
The integral of x/2 cos(5x) is 1/50 (5xsin(5x)+cos(5x))+C