What is the integral of xcos(182x) ?

The integral of xcos(182x) is 1/33124 (182xsin(182x)+cos(182x))+C

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integral of xcos(182x)

\int x cos (182 x) d x

\frac{1}{33124} (182 x sin (182 x) + cos (182 x)) + C

Solution steps

\int x cos (182 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{33124} d u

= \int \frac{u cos ( u )}{33124} d u

Take the constant out:

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

= \frac{1}{33124} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= \frac{1}{33124} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{33124} (u sin (u) - (- cos (u)))

Substitute back

u = 182 x

= \frac{1}{33124} (182 x sin (182 x) - (- cos (182 x)))

Simplify

= \frac{1}{33124} (182 x sin (182 x) + cos (182 x))

Add a constant to the solution

= \frac{1}{33124} (182 x sin (182 x) + cos (182 x)) + C

f (x) = sin (x) + 3, at x = π

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What is the integral of xcos(182x) ?
The integral of xcos(182x) is 1/33124 (182xsin(182x)+cos(182x))+C