What is the integral of 2xcos(5x) ?

The integral of 2xcos(5x) is 2/25 (5xsin(5x)+cos(5x))+C

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integral of 2xcos(5x)

\int 2 x cos (5 x) d x

\frac{2}{25} (5 x sin (5 x) + cos (5 x)) + C

Solution steps

\int 2 x cos (5 x) d x

Take the constant out:

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

= 2 \cdot \int x cos (5 x) d x

Apply u-substitution

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

= 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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = 5 x

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

Simplify

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

= \frac{2}{25} (5 x sin (5 x) + cos (5 x))

Add a constant to the solution

= \frac{2}{25} (5 x sin (5 x) + cos (5 x)) + C

\frac{\partial}{\partial x} (2 x^{4} y)

\frac{d y}{d x} = - a (y - b)

y = x^{3} - 4 x

\frac{2 - x}{e ^{3 x}}

What is the integral of 2xcos(5x) ?
The integral of 2xcos(5x) is 2/25 (5xsin(5x)+cos(5x))+C