What is the integral of xcos(5x) ?

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

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integral of xcos(5x)

\int x 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

\int_{0}^{1} \frac{1 + x}{1 + x ^{2}} d x

y = x^{5} sin^{2} (2 x)

\int_{- 2}^{2} \frac{1}{4 + x ^{2}} d x

x y^{^{'}} - y = 2 x ln (x)

f (x) = \frac{x ^{2}}{x + 3}, (3, \frac{3}{2})

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